Comparison of Methods for Computing Yearly Growth Rates from 

Weekly and Monthly Data, 1978 to 2005 

Carol Joyce Blumberg 

Energy Information Administration—Office of Oil and Gas

2035 Lyttonsville Road 

Silver Spring MD 20910-2264 USA 

Phone: (202) 586-6565 Fax: (202) 586-4913 

Email: carol.blumberg@eia.doe.gov 

Keywords: Growth Rates; Time Series; Petroleum; Establishment Surveys; Seasonality 

Abstract

The Energy Information Administration (EIA) collects weekly and monthly data on volumes for a variety of petroleum

products. Monthly data generally come from a census of appropriate companies. Weekly data are from samples of these

companies. Estimates of total volume are then formed from these data. 

The approximate release times of the estimates are: 

Monthly estimates based on weekly data (called Monthly-from-Weekly and abbreviated MFW)---11 days

Preliminary estimates from the monthly census---60 days 

Final reported volumes that include late submissions and resubmissions---September of the following year. 

The gold standard for computing yearly growth rates in this paper is the final reported volume for a particular time period

divided by the final reported volume for that same time period in the previous year. However, EIA customers, such as

financial analysts and industry experts, want estimates of yearly growth quickly. So, oftentimes, the gold standard is not

practical. 

This study focused on constructing practical alternatives to the gold standard. The questions investigated were:

1. If MFW or preliminary estimates are used as the numerators, what are the best denominators to use?

2. If cumulative volumes for 3-, 6-, or 9-months or an entire year based on MFW or preliminary estimates are used as 

numerators:

a. What are the best denominators?

b. Is there seasonality in the growth rates?

The criteria for deciding between the alternative ratios were the differences in means and standard deviations, mean square 

error, and correlations with the gold standard. Ratios were further compared on percentage of times they were within 1% and

2% or in the same direction (both positive or both negative) as the gold standard. Data from 1978 through 2005 were used. 

The focus of this paper is the total volume supplied of all grades and types of finished motor gasoline. Similar results were 

obtained when total volume for all types of distillate fuel oil was used. 

Although different methods did better on certain criteria, overwhelmingly the best denominator in all cases was the 

preliminary estimate. No seasonality or other patterns were found in the growth rates, even though there is seasonality

(periodicity of 12 months) in the monthly volumes used to compute the ratios. 

Introduction 

Statement of Purpose 

This paper is part of a bigger project to compare the congruence of the monthly volumes for finished motor gasoline 

supplied, distillate fuel oil supplied, and total product supplied as reported in various publications of the Petroleum Division

of the Energy Information Administration (EIA). The focus of this paper is to compare six methods of computing yearly

growth rates using data from 1978 to 2005 for finished motor gasoline supplied. The ideal ratio or gold standard (which will
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

also be called Method A) is formed using the final reported volumes from Petroleum Supply Annual (PSA) for a certain time

period in one year divided by the PSA reported volumes for the previous year during the same time period. 

Each of the growth rates methods, including Method A, is based on either monthly-based estimates and/or weekly-based

estimates. The monthly-based estimates used in computing these growth rates are the preliminary estimates from Petroleum 

Supply Monthly (PSM) and the final reported volumes from PSA. The PSA reported volumes are updates of the preliminary 

PSM values and include late submissions and resubmissions with corrections. 

Although there are multiple approaches for deriving monthly estimates from weekly data, based on earlier work (Blumberg,

2007; Kirkendall & Martin, 1993) it was decided to use only the monthly-from weekly (MFW) approach. The MFW

estimates are weighted means of the weeks that have any days within a particular month. Some of this earlier work, that will

be described further below, showed that, when yearly growth rates comparing a month from one year to that same month in a 

previous year, the differences are meaningfully insignificant between the different methods of computing monthly volumes 

from weekly data. The MFW was chosen because it is the one used in Weekly Petroleum Status Report (WPSR) and in PSM. 

As its title implies, WPSR is published weekly (usually on Wednesdays) and uses the weekly samples of companies as its

data source for product supplied volumes. A week is defined by EIA for product supplied volumes as starting on a Friday at 

7:01 a.m. and ending the next Friday at 7 a.m.. 

The reason that comparing growth rates based on weekly data, monthly data as reported in PSM, and monthly data as 

reported in PSA is important is that the weekly-based estimates for a month are available approximately 11 days of the end of

the month while the PSM and PSA data take longer to be released. The PSM data are available within 60 days of the end of

month. For example, the PSM that included data through March 2007 was released on May 30, 2007. The PSA values are 

released approximately nine months after the end of the calendar year that they cover. 

Since the weekly-based estimates are based on samples and the PSM and PSA measurements are based on censuses of the

same populations, the PSM and PSA data are generally more accurate, with the PSA measurements being more accurate than

the PSM measurements due to corrections made between the publication of the PSM values and the PSA values. So, the

question of main interest here is whether growth rates using weekly-based estimates and/or PSM values can mirror closely 

the growth rates computed using the PSA numbers as both the numerator and the denominator. 

This paper will be divided into five major parts. The first part is this introduction. The second part will summarize earlier

work done by the author when one-month periods were examined. The third part will investigate what methods best 

approximate Method A for computing yearly growth rates when three-month periods are used. The fourth part will

investigate what methods best approximate Method A when computing cumulative growth rates based on the first 3 months,

first 6 months, first 9 months, and/or an entire year are used. The fifth part takes some of the inconclusive results from the

second, third, and fourth parts and studies those results further to be able to make better conclusions. 

Definitions 

Product Supplied

Product Supplied is defined by EIA as equal to (Field Production + Refinery and Blender Net Production + Imports + 

Adjustments) – (Stock Change + Refinery and Blender Net Inputs + Exports). 

Monthly-from-Weekly Estimates 

The monthly-from-weekly (MFW) estimates are weighted averages of the weeks that contain the days of a certain month.

For example, for April 2007, the weighted average was {5*(data reported for 4/6/07) + 7*(data reported for 4/13/07) + 

7*(data reported for 4/20/07) + 7*(data reported for 4/27/06) + 4*(data reported for 5/4/07)}/30.

Gold Standard (Method A) 

This paper will take as the “gold standard” the year-to-year growth rate as defined by

A(Period ,t) =

⎜

⎜

⎛ Period PSA (t)

−1

⎟

⎟

⎞

*100% where Period is defined by the cumulative number of months under consideration 

⎝ Period PSA (t −1) ⎠

(either 1, 3, 6, 9 or 12 months for this paper) and t = number of years since 1978 and where PSA is the volume reported in

Petroleum Supply Annual. This method will be abbreviated throughout the paper as Method A. Method A will be compared

to the following five other methods of defining year-to-year growth rates.
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Method 1 

Method ⎛ Period (t) ⎞1 uses ⎜

PSM − 1⎟ *100% ⎜ ⎟ ⎝ Period PSA (t − 1) ⎠

method used by EIA’s Short-Term Energy Outlook (STEO) when preparing Figure 8 (U.S. Petroleum Products Consumption

Growth.) 

where PSM = the volume reported in Petroleum Supply Monthly. This is the 

Method 2 

Method 2 uses ⎜

⎜

⎛ Period PSM (t)

− 1

⎟

⎟

⎞

*100% . Method 2 is presently not used in any EIA publications. 

Period (t − 1) ⎝ PSM ⎠

Method 3 

Method 3 uses ⎜

⎜

⎛ Period MFW (t)

− 1

⎟

⎟

⎞

*100% . Method 3 is presently not used in any EIA publications. 

Period (t − 1) ⎝ MFW ⎠

Method 4 

Method ⎛ Period (t) ⎞4 uses MFW − 1 *100% . Method 4 is presently not used in any EIA publications, although a slight ⎜

⎜

⎟

⎟

⎝ Period PSM (t − 1) ⎠

variant of it is used in Table 1 of WPSR. 

Method 5 

ethod ⎛ ⎞ 5 uses Period (t)

⎜

MFW − 1⎟ *100% . Method 5 is presently not used in any EIA publications. ⎜

⎝

(

⎟ Period PSA t − 1) ⎠

M

Table 1 shows the relationships between the 6 methods. 

Table 1. Relationships Between the 6 Methods 

Numerator

Denominator PSA PSM MFW 

PSA Method A Method 1 Method 5

PSM Method 2 Method 4

MFW Method 3

Data Source 

The monthly values for the PSA, PSM, and MFW are regularly extracted from the publication files of the appropriate 

publications and put into an Excel file by colleagues at EIA. This Excel file was used as the data source for this paper. The 

Excel file is in thousand barrels per day. It was decided to convert the data to million barrels per month so that cumulative

volumes supplied over the different portions of the year would be easier to examine. It should be noted that the data in the 

Excel file are rounded to whole numbers. The conversion to million barrels per month caused some loss of precision, but this

loss was slightly less than if the means in thousand barrels per day over the 3-, 6-, 9-, and 12-month periods had been

computed. 

Descriptive Statistics for the Raw Data

Descriptive statistics for the raw data may be needed by the reader to help interpret some of the results in this paper. These

statistics are given in Tables 2 and 3. Table 2 is for 1- and 3-month periods. Table 3 is for 6-, 9-, and 12-month periods. The
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

major things to notice are that the means, medians, and standard deviations for PSA, PSM, and MFW are very close to each 

other for all sub-tables within Tables 2 and 3. Further, the standard deviations are close across all 3-month periods and

across both 6-month periods, even though the means change due to seasonal demand. Further, the PSA means are greater

than the PSM means and both are, in general, greater than the MFW means. Although not shown here, the distribution of the

raw data for each of the three measurements (PSA, PSM, and MFW) for all months over the 28 years (n = 336) is

approximately normally distributed. 

Table 2. Descriptive Statistics for Finished Motor Gasoline Supplied for One-Month and Three-Month Periods, 

1978-2005 (million barrels)

Measurement Mean 

Standard 

Deviation Median Minimum Maximum 

Percent of 

the Time 

Greater 

Than PSA 

Percent of 

the Time 

Greater 

Than PSM 

Percent of 

the Time 

Greater 

Than MFW 

All Months Individually (n=336) 

PSA 232.06 27.15 227.57 168.00 295.65 -- 62.20 61.90 

PSM 231.54 26.80 227.35 168.61 293.07 34.82 -- 55.36 

MFW 231.10 27.01 227.12 168.00 293.32 37.80 44.35 --

First 3 Months (January to March) (n=28) 

PSA 658.65 71.97 638.61 558.69 805.87 -- 75.00 64.29 

PSM 657.25 71.16 638.22 558.06 799.13 25.00 -- 60.71 

MFW 657.02 73.21 635.69 554.52 802.92 35.71 39.29 --

Second 3 Months (April to June) (n=28) 

PSA 706.17 73.13 686.60 608.58 842.63 -- 75.00 75.00 

PSM 703.83 71.88 684.24 608.15 842.27 25.00 -- 67.86 

MFW 699.13 73.39 678.16 583.83 849.18 25.00 32.14 --

Third 3 Months (July to September) (n=28)

PSA 717.27 78.18 694.43 610.30 858.24 -- 71.43 53.57 

PSM 715.75 76.80 693.33 610.02 852.60 28.57 -- 53.57 

MFW 716.64 75.52 701.02 606.50 848.70 46.43 46.43 --

Fourth 3 Months (October to December) (n=28) 

PSA 702.41 77.78 681.83 598.43 841.71 -- 75.00 57.14 

PSM 701.45 76.98 681.80 597.60 840.02 25.00 -- 53.57 

MFW 700.07 78.67 678.10 588.49 841.46 42.86 46.43 --

All 3 Month Periods Individually (n=112) 

PSA 696.13 77.60 681.70 558.69 858.24 -- 74.11 62.50 

PSM 694.57 76.57 681.62 558.06 852.60 25.89 -- 58.93 

MFW 693.21 77.43 678.16 554.52 849.18 37.50 41.07 --

Notes: 

(1) PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; MFW = Monthly estimates based on weekly data. 

(2) The complementary percentages in the “Percent of the Time Greater Than ...” columns for the “All Months Individually” portion of the 

table do not add to 100 percent since in a few instances the PSA, PSM or MFW monthly values were equal to each other. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3. Descriptive Statistics for Finished Motor Gasoline Supplied for Six-Month, Nine-Month,

and Twelve-Month Periods, 1978-2005 (million barrels) 

Percent of Percent of Percent of 

the Time the Time the Time 

Standard Greater Greater Greater 

Measurement Mean Deviation Median Minimum Maximum Than PSA Than PSM Than MFW 

First 6 Months (January to June) (n=28) 

PSA 1364.64 144.50 1319.75 1174.99 1643.89 -- 67.86 75.00 

PSM 1360.90 142.47 1317.44 1173.33 1639.14 32.14 -- 60.71 

MFW 1355.99 145.82 1311.12 1138.22 1651.90 25.00 39.29 --

Second 6 Months (July to December) (n=28) 

PSA 1419.68 155.91 1375.17 1209.16 1699.95 -- 75.00 67.86 

PSM 1417.20 153.70 1374.22 1207.62 1690.96 25.00 -- 57.14 

MFW 1416.71 153.78 1375.51 1194.99 1688.20 32.14 42.86 --

All 6 Month Periods Individually (n=56) 

PSA 1392.16 151.51 1353.15 1174.99 1699.95 -- 71.43 71.43 

PSM 1389.05 149.56 1350.55 1173.33 1690.96 28.57 -- 58.93 

MFW 1386.35 151.61 1342.05 1138.22 1688.20 28.57 41.07 --

First 9 Months (January to September) (n=28) 

PSA 2082.09 222.22 2014.35 1788.44 2500.91 -- 67.86 67.86 

PSM 2076.82 218.69 2010.52 1787.54 2491.93 32.14 -- 50.00 

MFW 2072.79 220.57 2009.97 1744.84 2498.84 32.14 50.00 --

Entire Year Total (January to December) (n=28) 

PSA 2784.07 299.69 2694.65 2386.46 3342.18 -- 78.57 78.57 

PSM 2777.83 295.31 2691.38 2385.06 3329.90 21.43 -- 60.71 

MFW 2772.40 298.60 2683.07 2332.90 3340.02 21.43 39.29 --

Note: 

PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; MFW = Monthly estimates based on weekly data. 

Evaluation Criteria for Comparing the Methods 

For Study 1, Study 2, and Study 3 the methods are compared first using the descriptive statistics of the mean, standard 

deviation, mean square error and correlations with the “gold standard” of Method A. Also, the growth rates given by Method

A and by the other methods are compared using paired (matched) t-tests. Next, the other methods are compared in terms of

the percentage of the time that they are within 1% and within 2% of Method A. Further, the other methods are compared to

Method A in terms of the percentage of the time that they give growth rates in the same direction (that is, both give positive 

growth rates or both give negative growth rates) as Method A. For Study 4, the above criteria will be used to compare 

directly to each other the methods that looked most promising from the first three studies. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

STUDY 1 

Comparison of Growth Rates Based on One-Month Periods 

Study 1 was a preliminary study and used data only from 1995 through 2004. It compared 15 methods of using the monthly 

data from PSA and PSM and the weekly data from WPSR to form year-to-year growth rates. In addition to the gold standard 

of Method A and the five methods described in the introduction to this paper, growth rate ratios were also formed using two 

other approaches of computing monthly estimates from weekly data. These other two approaches were (i) computing the 

mean of the four weeks preceding the last Friday of the month and (ii) computing the mean of the four weeks that contain the

most days of the month. When the values computed using these two other approaches replaced the MFW values it was found 

that there were no statistically significant or meaningfully significant differences between the use of MFW and these other 

two approaches. So, it was decided to use only MFW to compute monthly estimates from weekly numbers in Study 2, Study

3, and Study 4. Further, some methods were tried that used PSM values as the denominator for the first six months of the year

and PSA values as the denominator for the last six months of the year. These “hybrid” methods did not perform well. Hence,

it was decided not to look at these “hybrid” methods in further investigations. See Blumberg (2007) for details. 

Table 4 provides a summary of the comparisons of Methods 1 to 5 to Method A when one-month periods are used. It was

found that Method 2, that has PSM for the present year in the numerator and PSM for the previous year in the denominator, is 

the best method to use to approximate growth rates as calculated by Method A. It has descriptive statistics that are the closest

of all 14 methods to Method A. Its correlation with Method A is .905. Further, its year-to-year growth rates are within 2% of

Method A more often than other methods. Finally, far less of the time, it gives a different direction for year-to-year growth

rates than Method A does as compared to the other methods (except for Method 1.) It can also be seen that Method 3 does 

the worst and that Methods 4 and 5 do about the same as compared to Method A. 

STUDY 2 

Comparison of Growth Rates Based on Three-Month Periods 

For this study, quarters are defined in the usual way as: first quarter = January to March; second quarter = April to June; third 

quarter = July to September; and fourth quarter = October to December. The 1978 data were used only in the denominators

for the 1979 growth rate ratios. 

Descriptive Statistics 

Table 5 gives the means, standard deviations, and correlations with Method A along with the other major evaluation criteria

statistics for the six growth rate methods. Table 6 gives more details about the descriptive statistics, including means,

standard deviations, medians, minimums, maximums, and ranges of the growth rate methods. 

The methods that use PSM as the numerator have high correlations with Method A of .982 (Method 2) and .986 (Method 1)

and their standard deviations, minimums, maximums and ranges are extremely close to that of Method A (see Table 5.)

Method 1 (PSM/PSA), however, has a lower mean than Method A in all four quarters separately and combined (see Table 6.) 

Method 2 (PSM/PSM) has means fairly close to Method A. 

The three methods that use MFW as the numerator have good correlations with Method A between .802 and .877 (R2

between .643 and .769) and their standard deviations are close to that of Method A for all quarters separately and combined. 

Method 3 (MFW/MFW) has a higher mean overall than Method A. It also has higher means for each quarter separately,

except for the third quarters where Method 3’s mean is lower than Method A’s. Method 4 (MFW/PSM) and Method 5 

(MFW/PSA) have lower means overall than Method A and this holds true for all quarters separately, except for third quarters 

where Method 4 has a lower mean than Method A and Method 5 has a higher mean than Method A. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4. Major Statistics Used to Evaluate the Year-to-Year Growth Rate Methods for Finished Motor Gasoline

 Supplied when Monthly Periods are Used, 1995-2004 

Percent Percent 

Corre- Differ- of Time of Time 

lation 

with

ence in 

Mean 

p-value

for the

Within 

1% of

Within 

2% of

Percent 

of Time 

Method Method in Standard Method from Differ- Method Method in Same 

Number Symbols Mean Deviation A Method A ence A A Direction 

A PSA/PSA 1.846 1.892 -- -- -- -- -- --

1 PSM/PSA 1.442 1.974 0.941 0.404 <.0001 83.33 96.67 95.00 

2 PSM/PSM 1.819 1.907 0.905 0.028 0.716 80.83 97.50 92.50 

3 MFW/MFW 1.852 2.155 0.438 -0.005 0.978 34.67 69.17 76.67 

4 MFW/PSM 1.840 2.137 0.665 0.006 0.971 44.17 75.00 85.00 

5 MFW/PSA 1.463 2.154 0.721 0.383 0.007 44.17 78.33 83.33 

Notes: 

(1) All statistics in this table are based on 120 observations (12 months for each of the years 1995 to 2004.) 

(2) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular month 

in a particular year for the measure listed and the symbols in the denominator represent the value from the same month in the 

previous year. PSA = Petroleum Supply Annual; PSM =Petroleum Supply Monthly; MFW = Monthly estimates based on

weekly data. 

(3) The values in the third, fourth, and sixth columns are year-to-year growth rates expressed as percentages. For example, 

the mean year-to-year growth rate using Method A for the 120 months from January 1995 to December 2004 was 1.846

percent and the standard deviation was 1.892 percent. 

(4) The symbol -- in the Method A row represents “Not Applicable”. 

(5) The correlations reported here are Pearson Product-Moment correlations of the year-to-year growth rates determined by

Methods 1 to 5 with the year-to-year growth rates for Method A. All correlations have a probability value (p-value) of 

<.0001. 

(6) The percentages reported in the sixth column are mean differences in year-to-year growth rates. For example, the mean 

difference between Method A for year-to-year growth rate for the 120 months and the PSM/PSM ratio (Method 2) for yearto-year growth rate for the 120 months is 0.028 percent (that is, Method A’s mean is 0.028 percent higher.) 

(8) The percentages reported in the eighth and ninth columns are the percentage of times that each method gives an answer

within 1% (eighth column) or 2% (ninth column) of the ratio defined by Method A. For example, for Method 2, 80.83

percent of the time it gives a year-to-year growth rates within 1% of the growth rate given by Method A and 97.50 percent of

the time it is within 2% of the Method A growth rate. 

(9) The last column gives the percentage of times that each method gives a growth rate in the same direction as Method A.

That is, it is the percentage of the time that each method and Method A both give positive growth rates or both give negative

growth rates. For example, 92.50 percent of the time Method 2 and Method A are either both positive or both negative. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5. Major Statistics Used to Evaluate the Year-to-Year Growth Rate Methods for Finished Motor Gasoline 

Supplied Based on Quarterly Sums, 1979 to 2005 

Method 

Number 

Method in 

Symbols Mean 

Standard 

Deviation 

Pearson 

Correlation 

with

Method A

Difference 

in Mean

from

Method A

p-value for 

the 

Difference 

Percent of 

the Time 

Within 

2% of

Method A

Percent 

of the

Time in 

the Same

Direction 

A PSA/PSA 0.823 2.549 -- -- -- -- --

1 PSM/PSA 0.600 2.525 0.986 0.223 <.0001 100.00 94.44 

2 PSM/PSM 0.808 2.592 0.982 0.015 0.7526 100.00 96.30 

3 MFW/MFW 0.837 2.865 0.802 -0.015 0.9302 83.33 80.56 

4 MFW/PSM 0.599 2.640 0.877 0.224 0.0742 90.74 87.96 

5 MFW/PSA 0.391 2.579 0.876 0.432 0.0006 90.74 85.19 

Notes: 

(1) All statistics in this table are based on 108 observations (four quarters for each of the years 1979 to 2005.) 

(2) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular quarter 

in a particular year for the measure listed and the symbols in the denominator represent the value from the same quarter in the

previous year. PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based

on weekly data. 

(3) The values in the third, fourth, and sixth columns are year-to-year growth rates expressed as percentages. For example, 

Method A over the 108 quarters from 1979 to 2005 has a mean ratio of 0.823 percent and a standard deviation of 2.549

percent. 

(4) The symbol -- in the Method A row represents “Not Applicable”. 

(5) The correlations reported here are Pearson Product-Moment correlations of the year-to-year growth rates determined by

Methods 1 to 5 with the year-to-year growth rates for Method A. All correlations have a probability value (p-value) of 

<.0001. 

(6) The sixth column is mean differences in year-to-year growth rates. For example, the mean difference between Method A 

and Method 2 for the 108 quarters is 0.015 percent (that is, Method A’s mean is 0.015 percent higher.) 

(7) The percentages reported in the eighth column are the percentage of times that each method gives a growth rate within 2%

of the growth rate given by Method A. For example, for Method 3, 83.33 percent of the time it is within 2% of the Method A 

growth rate. 

(8) The last column gives the percentage of the time that each method gives a growth rate in the same direction as Method A. 

That is, it is the percentage of the time that each method and Method A both give positive growth rates or both give negative

growth rates. For example, 80.56 percent of the time Method 3 and Method A are either both positive or both negative. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 6. Descriptive Statistics for the Year-to-Year Growth Rates (as Percentages) for Finished

Motor Gasoline Supplied, 1979-2005 

Method in Standard 

Method Number Symbols Mean Deviation Median Minimum Maximum Range 

All Quarters (n = 108) 

A PSA/PSA 0.823 2.549 1.420 -8.320 5.173 13.493 

1 PSM/PSA 0.600 2.525 1.079 -8.566 4.459 13.025 

2 PSM/PSM 0.808 2.592 1.326 -8.693 5.293 13.986 

3 MFW/MFW 0.837 2.865 1.461 -7.899 5.378 13.277 

4 MFW/PSM 0.599 2.640 1.381 -9.214 5.093 14.307 

5 MFW/PSA 0.391 2.579 0.776 -9.088 5.058 14.146 

First Quarters (n = 27) 

A PSA/PSA 0.960 2.780 1.845 -8.320 5.173 13.493 

1 PSM/PSA 0.744 2.675 1.303 -8.258 4.364 12.622 

2 PSM/PSM 0.946 2.752 1.387 -8.550 5.293 13.843 

3 MFW/MFW 1.004 3.140 1.318 -7.377 5.378 12.755 

4 MFW/PSM 0.892 2.689 1.396 -5.604 4.244 9.848 

5 MFW/PSA 0.692 2.650 0.412 -5.303 4.270 9.573 

Second Quarters (n = 27) 

A PSA/PSA 0.757 2.456 1.062 -6.042 4.169 10.211 

1 PSM/PSA 0.422 2.392 0.625 -6.047 4.202 10.249 

2 PSM/PSM 0.754 2.514 0.817 -6.112 4.897 11.009 

3 MFW/MFW 0.809 2.911 1.575 -6.385 5.008 11.393 

4 MFW/PSM 0.046 2.695 0.884 -6.729 3.987 10.716 

5 MFW/PSA -0.286 2.527 0.314 -6.664 3.619 10.283 

Third Quarters (n = 27) 

A PSA/PSA 0.785 2.485 1.110 -7.590 4.100 11.690 

1 PSM/PSA 0.580 2.515 0.929 -7.822 4.459 12.281 

2 PSM/PSM 0.760 2.540 1.036 -7.862 4.494 12.356 

3 MFW/MFW 0.695 2.823 1.435 -7.899 5.163 13.062 

4 MFW/PSM 0.859 2.378 1.429 -6.746 5.093 11.839 

5 MFW/PSA 0.680 2.374 1.334 -6.705 5.058 11.763 

Fourth Quarters (n = 27) 

A PSA/PSA 0.789 2.603 1.550 -8.307 3.974 12.281

1 PSM/PSA 0.652 2.640 1.350 -8.566 3.485 12.051 

2 PSM/PSM 0.771 2.697 1.489 -8.693 3.500 12.193 

3 MFW/MFW 0.841 2.726 1.487 -7.842 4.307 12.149 

4 MFW/PSM 0.598 2.840 1.467 -9.214 4.118 13.332 

5 MFW/PSA 0.478 2.769 1.248 -9.088 4.099 13.187 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Notes for Table 6: 

(1) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular quarter 

in a particular year for the measure listed and the symbols in the denominator represent the value from the same quarter in the

previous year. PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based

on weekly data. 

(2) The values in the third through eighth columns are year-to-year growth rates expressed as percentages. For example, 

Method A over the 108 quarters from 1979 to 2005 has a mean ratio of 0.823 percent and a standard deviation of 2.549

percent. 

(3) Totals across components, where applicable, may not add due to independent rounding. 

Effects of Outliers on the Results

The effects of outliers were considered carefully, since all of the methods being compared use ratio estimators. A growth rate

that looks like a potential outlier may be caused by having a numerator or denominator that is larger or smaller than expected

for the period under consideration. Sometimes the individual numerator and denominator may be outliers, but their ratio may 

not be an outlier. It was found that there were some outliers, but that these outliers had no meaningful effects on the results. 

The main effect of the outliers was to increase the standard deviations for all the methods. This actually made it easier to see 

patterns. 

Significance Testing of Mean Differences in Growth Rates 

The statistical tests for deciding whether there were significant differences between the mean growth rate using Method A 

and the other five methods are given in Table 7. The differences in means and the p-values are also given in Table 5. Paired 

(matched) t-tests were used since the growth rates are indexed by year and quarter. It needs to be remembered that growth 

rate differences, and not raw differences, are being tested here. 

For the methods that use PSM as the numerator, Method 1’s (PSM/PSA) mean difference from Method A is significant while

Method 2’s (PSM/PSM) mean difference from Method A is not significant across all quarters. However, the standard

deviation of the differences is smaller for Method 1 than for Method 2. To see the combined influence of the bias (mean 

difference from Method A) and the variability (variance of the differences), the mean square error, defined by (bias)2

+ 

(standard deviation of the differences)2 was computed. Based on the mean square errors, it is a tie between Method 1 and 

Method 2 at this point. 

For the methods that use MFW as the numerator, Method 3 (MFW/MFW) has less bias than the Method 4 (MFW/PSM) and

Method 5 (MFW/PSA). But, Methods 4 and 5 have smaller standard deviations of the differences between them and Method

A. So, to see the combined influence, mean square errors were computed. Based on the mean square errors, Method 3 is

inferior to Method 4 and Method 5. However, the mean square errors are close for Method 4 and Method 5 for all quarters 

combined and for the four quarters separately. 

Percentage of the Time Within 1% and 2% of the Method A Growth Rate 

From Table 8 it can be seen that over all 108 quarters that 95.37 percent of the time the growth rates defined by Method 1 

(PSM/PSA) or Method 2 (PSM/PSM) were within 1% of Method A and 100.00 percent of the time these two methods had 

growth rates within 2% of Method A. When MFW was used as the numerator, over all 108 quarters, Method 3 (MFW/MFW) 

was inferior to Method 4 (MFW/PSM) and Method 5 (MFW/PSA). Method 3 was within 1% and 2% of Method A, 42.59

percent and 83.33 percent of the time, respectively. Methods 4 and 5 were tied and were within 1% and 2% of Method A,

58.33 percent and 90.74 percent of the time, respectively. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 7. Differences for Finished Motor Gasoline Supplied Between Method A and the Other Growth Rate Methods,

1979-2005

Standard 

Largest Largest Deviation of Mean 

Method Method in Under- Over- Mean the Square 

Number Symbols estimate estimate Difference Differences t p-value Error 

All 108 Quarters 

1 PSM/PSA -1.907 0.623 0.223 0.418 -5.546 <.0001 0.224 

2 PSM/PSM -1.759 1.890 0.015 0.489 -0.316 0.7526 0.239 

3 MFW/MFW -6.126 6.442 -0.015 1.731 0.088 0.9302 2.996 

4 MFW/PSM -5.391 2.716 0.224 1.292 -1.803 0.0742 1.718 

5 MFW/PSA -5.631 3.017 0.432 1.275 -3.518 0.0006 1.813 

First Quarters (n =27) 

1 PSM/PSA -1.259 0.623 0.216 0.413 -2.712 0.0117 0.217 

2 PSM/PSM -1.237 0.766 0.014 0.541 -0.136 0.8925 0.292 

3 MFW/MFW -4.793 2.350 -0.044 1.705 0.134 0.8947 2.910 

4 MFW/PSM -2.520 2.716 0.068 1.261 -0.280 0.7815 1.596 

5 MFW/PSA -2.675 3.017 0.268 1.297 -1.075 0.2921 1.754 

Second Quarters (n = 27) 

1 PSM/PSA -1.907 0.226 0.335 0.579 -3.009 0.0058 0.448 

2 PSM/PSM -1.759 1.890 0.003 0.622 -0.024 0.9811 0.387 

3 MFW/MFW -6.126 6.442 -0.052 2.208 0.122 0.9037 4.880 

4 MFW/PSM -5.391 1.457 0.712 1.522 -2.430 0.0223 2.824 

5 MFW/PSA -5.631 0.873 1.043 1.408 -3.850 0.0007 3.071 

Third Quarters (n = 27) 

1 PSM/PSA -1.173 0.359 0.204 0.353 -3.010 0.0057 0.166 

2 PSM/PSM -0.746 0.975 0.025 0.425 -0.304 0.7632 0.181 

3 MFW/MFW -3.220 4.139 0.090 1.700 -0.275 0.7856 2.897 

4 MFW/PSM -3.536 2.208 -0.074 1.211 0.319 0.7523 1.473 

5 MFW/PSA -3.828 1.966 0.105 1.231 -0.442 0.6618 1.527 

Fourth Quarters (n = 27) 

1 PSM/PSA -0.924 0.389 0.137 0.258 -2.746 0.0108 0.085 

2 PSM/PSM -0.844 0.602 0.018 0.352 -0.259 0.7980 0.124 

3 MFW/MFW -2.646 2.693 -0.053 1.281 0.213 0.8327 1.643 

4 MFW/PSM -2.027 1.729 0.191 1.060 -0.937 0.3575 1.160 

5 MFW/PSA -2.050 1.816 0.310 0.985 -1.636 0.1138 1.067 

Notes: 

(1) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular quarter in a particular

year for the measure listed and the symbols in the denominator represent the value from the same quarter in the previous year. PSA = 

Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based on weekly data. 

(2) The units for Largest Underestimate, Largest Overestimate, Mean Difference, and Standard Deviation of the Difference are percent.

The units for Mean Square Error are percent squared. 

⎛ Quarter (t) ⎞

(3) The method that Methods 1 to 5 are being compared to is Method A: A(Quarter ,t) =

PSA − 1 *100% . The percentages ⎜

⎜

⎟

⎟

Quarter (t − 1) ⎝ PSA ⎠

reported are differences in year-to-year growth rates. For example, the mean difference for year-to-year growth rates for the 108 quarters

between Method A and Method 1 (PSM/PSA) is 0.223 percent (that is, Method A’s mean is 0.223 percent higher.) 

(4) Mean Square Error = (bias)2

+ (standard deviation)2. Here, the bias = Mean Difference (the fifth column) and the standard deviation =

Standard Deviation of the Differences (the sixth column.) 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 8. Percentage of Time Each Year-to-Year Growth Rate Method is Within 1% and 2% of

Method A for Finished Motor Gasoline Supplied, 1979-2005

Percent 

Percent Percent Percent Percent Percent Percent of the

of the of the of the of the of the of the Time 

Time Time Time Time Time Time Not 

Method Method in Within Within Within Within Within Within Within 

Number Symbols -1% +1% -2% +2% 1% 2% 2%

All 108 Quarters 

1 PSM/PSA 69.44 25.93 74.07 25.93 95.37 100.00 0.00 

2 PSM/PSM 46.30 49.07 50.00 50.00 95.37 100.00 0.00 

3 MFW/MFW 17.59 25.00 37.96 45.37 42.59 83.33 16.67 

4 MFW/PSM 29.63 28.70 48.15 42.59 58.33 90.74 9.26 

5 MFW/PSA 29.63 28.70 53.70 37.04 58.33 90.74 9.26 

First Quarters (n =27) 

1 PSM/PSA 70.37 25.93 74.07 25.93 96.30 100.00 0.00 

2 PSM/PSM 29.63 59.26 40.74 59.26 88.89 100.00 0.00 

3 MFW/MFW 11.11 22.22 29.63 48.15 33.33 77.78 22.22 

4 MFW/PSM 18.52 33.33 44.44 48.15 51.85 92.59 7.41 

5 MFW/PSA 33.33 22.22 55.56 33.33 55.56 88.89 11.11 

Second Quarters (n = 27) 

1 PSM/PSA 62.96 25.93 74.07 25.93 88.89 100.00 0.00 

2 PSM/PSM 51.85 40.74 55.56 44.44 92.59 100.00 0.00 

3 MFW/MFW 25.93 14.81 44.44 37.04 40.74 81.48 18.52 

4 MFW/PSM 33.33 14.81 55.56 29.63 48.15 85.19 14.81 

5 MFW/PSA 22.22 25.93 55.56 25.93 48.15 81.48 18.52 

Third Quarters (n = 27) 

1 PSM/PSA 66.67 29.63 70.37 29.63 96.30 100.00 0.00 

2 PSM/PSM 55.56 44.44 55.56 44.44 100.00 100.00 0.00 

3 MFW/MFW 11.11 22.22 44.44 40.74 33.33 85.19 14.81 

4 MFW/PSM 40.74 33.33 44.44 48.15 74.07 92.59 7.41 

5 MFW/PSA 37.04 29.63 51.85 40.74 66.67 92.59 7.41 

Fourth Quarters (n = 27) 

1 PSM/PSA 77.78 22.22 77.78 22.22 100.00 100.00 0.00 

2 PSM/PSM 48.15 51.85 48.15 51.85 100.00 100.00 0.00 

3 MFW/MFW 22.22 33.33 33.33 55.56 55.56 88.89 11.11 

4 MFW/PSM 25.93 33.33 48.15 44.44 59.26 92.59 7.41 

5 MFW/PSA 25.93 37.04 51.85 44.44 62.96 96.30 3.70 

Notes: 

(1) To interpret the “Method in symbols” column, the symbols in the numerator represent the value from a particular quarter in a particular 

year and the symbols in the denominator represent the value from the same quarter in the previous year. PSA = Petroleum Supply Annual;

PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based on weekly data. 

(2) The percentages reported are the percentage of times that each method gives an answer within 1% or 2% (depending on the column) of

Method A of A(Quarter ,t) =

⎛

⎜

⎜

Quarter PSA (t)

− 1

⎞

⎟

⎟ *100% . For example, the first two entries for the Method 3 row for all 108 quarters 

Quarter (t − 1) ⎝ PSA ⎠

mean that 17.59 percent of the MFW/MFW ratios are below the Method A ratios, but are still within 1% of them and that 25.00 percent of

the MFW/MFW ratios are above the Method A ratios, but are still within 1% of them. 

(3) Totals across components, where applicable, may not add to 100 percent due to independent rounding. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Occurrence of Reported Wrong Directions of Growth Rates 

Sometimes for year-to-year growth rates it occurs that some methods yield a positive growth rate and other methods yield a 

negative growth rate for the same time period. This will occur especially when the growth rates are small (near 0 percent.) 

While the growth rates calculated by two different methods may be close to each other in their values, the fact that one is 

positive and one is negative is bothersome psychologically to the users of EIA data on volumes. This discrepancy in terms of

sign is described in Table 9, where the “gold standard” of Method A was used and all other methods were compared to it. 

Because there are only 27 years of data, reporting the results separately for each quarter would be misleading.

As seen in Tables 5 and 9, Method 1 (PSM/PSA) and Method 2 (PSM/PSM) give the same direction as Method A does 94.44

percent and 96.30 percent of the time, respectively. Methods 3 (MFW/MFW) is not as good on this criterion. It gives the

same direction as Method A does 80.56 percent of the time. Method 4 (MFW/PSM) and Method 5 (MFW/PSA) are in the 

middle. They give the same direction as Method A does 87.96 percent and 85.19 percent of the time, respectively. 

Table 9. Occurrence of Reported Correct and Wrong Directions of Growth Rates for Finished Motor Gasoline, 

1979 to 2005

Percent of the

Time that 

Method A and 

Percent of the Percent of the Percent of the the Other

Percent of the Time Both Time Method A Time Method A Method are in 

Method 

Number 

Method in 

Symbols 

Time Both 

Methods are Up 

Methods are 

Down

is Up and the 

Other is Down 

is Down and the 

Other is Up 

the Same

Direction 

1 PSM/PSA 72.22 22.22 3.70 1.85 94.44 

2 PSM/PSM 74.07 22.22 1.85 1.85 96.30 

3 MFW/MFW 64.81 15.74 11.11 8.33 80.56 

4 MFW/PSM 67.59 20.37 9.26 2.78 87.96 

5 MFW/PSA 62.04 23.15 13.89 0.93 85.19 

Notes: 

(1) All statistics in this table are based on 108 observations (4 quarters for each of the years 1979 to 2005.) 

(2) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular quarter 

in a particular year for the measure listed and the symbols in the denominator represent the value from the same quarter in the

previous year. PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based 

on weekly data. 

(3) The percentages reported are percentage of times that each method gives an answer within 1% or 2% (depending on the 

⎛ Quarter (t) ⎞

column) of Method A of PSA − 1 *100% . For example, the first two entries for the Method 2 row mean that ⎜

⎜

⎟

⎟

Quarter (t − 1) ⎝ PSA ⎠

74.07 percent of the PSM/PSM ratios and Method A ratios are both positive (that is, indicate upward growth rates) and that

22.22 percent of the PSM/PSM ratios and Method A ratios are both negative (that is, indicate downward growth rates.)

(4) Totals across components, where applicable, may not add due to independent rounding. 

Seasonality/Cyclic Patterns

In Tables 5 to 8, the results are given for all 108 quarters and for first quarters, second quarters, etc. separately. Although 

there are some minor perturbations, overall for all methods there is very little difference between the quarters. What 

differences do occur are not meaningful when looking at the big picture. To verify this statistically, a two-way ANOVA

(analysis of variance) was done using the Methods and the Quarters as the main factors (see Table 10.)
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 10. Analyses of Variance of Differences between Methods and Quarters, 1979-2005

Sum of Mean

Source of Variation df Squares Square F p-value 

Method 5 102.3893 20.4779 0.62 0.6817 

Quarters 3 151.4417 50.4806 1.54 0.2035 

Method by Quarter Interaction 15 24.8223 1.6548 0.05 >0.99995 

Error 624 20483.0221 32.8254 

Total 647 20761.6754 

Note: The six methods used in this analysis of variance are Method A and Methods 1 through 5. The four quarters used are 

January to March, April to June, July to September, and October to December. 

As can be seen in Table 10, there are no significant differences for either of the main effect of Quarter or for the Method by

Quarter interaction. The non-statistically significant result for Method is not in contradiction with the results discussed

earlier in this paper. This non-significance is caused by the high correlations of the growth rates between the Methods and 

because of the relatively high within-cell (Method by Quarter) standard deviations (see Table 5) as compared to the 

differences in the means between the methods. 

Time-series analyses using a multiple linear regression viewpoint (in the EViews software, Version 5.1) for both months and 

quarters as the base unit were also performed on the growth rates generated using Method A (the gold standard) as the

dependent variable and time (in months or quarter starting January 1982) and indicator variables for the months/quarters as

independent variables. The models showed neither a linear time trend nor any seasonality or other cyclic patterns. The 

regression results are in Table 11 (for months) and Table 12 (for quarters.) Method A growth rates from 1979 to 1981 were

not used because the growth rates during that period were quite volatile, as compared to the period of 1982 to 2005, due to a

gasoline shortage in the U.S. and other factors. Hence, it can be concluded that there is no significant seasonality in growth

rates even though the monthly volumes show high seasonality (see Figure 1.) 

Table 11: Regression Equation Results for Estimating Growth Rates for Method A from Months, 1982 to 2005.

Independent Variables Coefficient Standard Error t p-value 

Intercept (Constant) 0.009492 0.005887 1.612426 0.1080

Linear Trend 0.000026 0.000017 1.499839 0.1348

January -0.000420 0.006995 -0.059987 0.9522

February 0.000617 0.006994 0.088245 0.9297

March 0.001218 0.006994 0.174182 0.8619

April -0.000190 0.006993 -0.027135 0.9784

May 0.000243 0.006993 0.034698 0.9723

June -0.001777 0.006993 -0.254142 0.7996

July 0.000097 0.006993 0.013919 0.9889

August 0.001300 0.006992 0.185893 0.8527

September -0.001698 0.006992 -0.242794 0.8083

October -0.000747 0.006992 -0.106801 0.9150

November 0.000974 0.006992 0.139286 0.8893

R

2 = .009757 F(12, 275) = .225803 p = .997091

Notes: 

⎛ Month (t) ⎞

(1) The dependent variable is the growth rates as computed using Method A of A(Month ,t) =

PSA − 1 *100% .

⎜

⎜

⎟

⎟

⎝ Month PSA (t − 1) ⎠

(2) The coding used for the Linear Trend variable is 1=January 1982, 2=February 1982, ... 13=January 1983, ... 288=December 2005. 

(3) The variables of January, February, ... November are indicator variables. For example, the variable of February is coded 1 if the 

observed month was a February and is coded 0 otherwise. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 12: Regression Equation Results for Estimating Growth Rates for Method A from Quarters, 1982 to 2005. 

Independent Variables Coefficient Standard Error t p-value

Intercept (Constant) 0.944620 0.495789 1.905289 0.0599

Linear Trend 0.007613 0.005964 1.276514 0.2050

Quarter 1 0.040626 0.467416 0.086916 0.9309

Quarter 2 -0.062620 0.467225 -0.134026 0.8937

Quarter 3 -0.017006 0.467111 -0.036407 0.9710

R

2

= .018096 F(11, 84) = .419267 p = .794359 

Notes: 

⎛ Quarter (t) ⎞

(1) The dependent variable is the growth rates as computed using Method A of A(Quarter ,t) =

⎜

⎜

PSA − 1

⎟

⎟ *100% . 

Quarter (t − 1) ⎝ PSA ⎠

(2) The quarters are defined so that: first quarter is January, February and March; the second quarter is April, May and June; 

the third quarter is July, August, and September; and the fourth quarter is October, November, and December. 

(3) The coding used for the Linear Trend variable is 1=first quarter 1982, 2=second quarter 1982, ... 5=first quarter 1983, ... 

96=fourth quarter 2005. 

(4) The variables of Quarter 1, Quarter 2, and Quarter 3 are indicator variables. For example, the variable of Quarter 2 is

coded 1 if the observed quarter is the second quarter and is coded 0 otherwise. 

Figure 1. Petroleum Supply Annual (PSA) Volumes of Finished Motor Gasoline Supplied, 1982 to 2005

160

180

200

220

240

260

280

300

Million Barrels Per Month

198201198209198305198401198409198505198601198609198705198801198809198905199001199009199105199201199209199305199401199409199505199601199609199705199801199809199905200001200009200105200201200209200305200401200409200505

Month
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Main Results from Study 2 

• There does not appear to be any seasonality or other cyclic patterns for the growth rates.

• Using PSM as the numerator is much better than using MFW as the numerator.

• If the PSM measurement is used as the numerator, there is very little difference in all of the statistics for Method 1 

(PSM/PSA) and Method 2 (PSM/PSM). However, when combined with the results of the previous study (Blumberg, 2007)

and the results of Study 3, a more definite conclusion can be made that Method 2 is preferable to Method 1.

• If the MFW measurement is used as the numerator, then using MFW as the denominator (Method 3) is inferior to both

Method 4 (MFW/PSM) and Method 5 (MFW/PSA).

• No conclusion can be drawn from this study as to whether Method 4 or Method 5 is superior. 

STUDY 3 

Comparison of Cumulative Growth Rates Based on Different Length Time Periods 

The purpose of this study was to see which of Methods 1 to 5 are best as compared to the gold-standard of Method A 

(PSA/PSA) when the volume of barrels supplied is accumulated for the first three months, first six months, first nine months

or the entire year. 

Descriptive Statistics 

Table 13 gives the means, standard deviations, and correlations with Method A (along with the other major evaluation criteria 

statistics) for these time periods. Table 14 gives more details about the descriptive statistics, including means, standard 

deviations, medians, minimums, maximums, and ranges. 

The methods that use PSM as the numerator (Method 1 of PSM/PSA and Method 2 of PSM/PSM) have high correlations with

Method A of between .978 and .991 for all four time periods. For all time periods, the means for Method 2 are very close to 

those of Method A, while Method 1’s means are lower than Method A’s means. 

For the 3-, 6-, and 9-month periods Method 3 (MFW/MFW) has correlations with Method A between .830 and .841 (R2

between .689 and .707) while Method 4 (MFW/PSM) and Method 5 (MFW/PSA) have correlations with Method A between 

.878 and .894 (R2

between .770 and .798.) When the volumes supplied are summed over the entire year, the correlations are 

slightly higher at .890 (for Method 3), .926 (for Method 4) and .932 (for Method 5.) For all time periods, the means for

Method 3 are very close to those of Method A while those of Methods 4 and 5 are lower, with those of Method 5 being

consistently the lowest. These differences in means will be discussed in more detail in the section on significance testing.

The standard deviations are very close for Method A and all 5 methods within each time period, with the standard deviations

being higher for the first 3 months then for the 6-month, 9-month, and entire year time periods. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 13. Major Statistics Used to Evaluate the Year-to-Year Growth Rate Methods for Finished Motor Gasoline, 

1979 to 2005

DifferPearson ence in Percent Percent 

Correlation 

Mean 

from

p-value

of the Mean 

of the

Time 

of the

Time in 

Method 

Number 

Method in 

Symbols Mean 

Standard 

Deviation 

with

Method A

Method 

A 

Differe

nce 

Square 

Error 

Within 

2% of A

the Same

Direction 

First 3 Months 

A PSA/PSA 0.960 2.780 -- -- -- -- -- --

1 PSM/PSA 0.744 2.675 0.989 0.216 0.0117 0.217 100.00 96.30 

2 PSM/PSM 0.946 2.752 0.981 0.014 0.8925 0.292 100.00 96.30 

3 MFW/MFW 1.004 3.140 0.841 -0.044 0.8947 2.910 77.78 81.48 

4 MFW/PSM 0.892 2.689 0.894 0.068 0.7815 1.596 92.59 88.89 

5 MFW/PSA 0.692 2.650 0.887 0.268 0.2921 1.754 88.89 96.30 

First 6 Months

A PSA/PSA 0.847 2.255 -- -- -- -- -- --

1 PSM/PSA 0.569 2.136 0.984 0.277 0.0016 0.245 100.00 96.30 

2 PSM/PSM 0.837 2.229 0.978 0.009 0.9185 0.222 100.00 96.30 

3 MFW/MFW 0.889 2.482 0.830 -0.042 0.8778 1.956 81.48 81.48 

4 MFW/PSM 0.447 2.240 0.878 0.400 0.0729 1.397 88.89 92.59 

5 MFW/PSA 0.179 2.128 0.887 0.668 0.0028 1.546 88.89 85.19 

First 9 Months

A PSA/PSA 0.821 2.164 -- -- -- -- -- --

1 PSM/PSA 0.569 2.081 0.987 0.252 0.0011 0.192 100.00 92.59 

2 PSM/PSM 0.806 2.143 0.981 0.015 0.8523 0.177 100.00 100.00 

3 MFW/MFW 0.815 2.346 0.835 0.006 0.9807 1.707 85.18 96.30 

4 MFW/PSM 0.583 2.025 0.886 0.239 0.2308 1.078 92.59 96.30 

5 MFW/PSA 0.346 1.954 0.893 0.475 0.0177 1.178 92.59 96.30 

Entire Year

A PSA/PSA 0.812 2.203 -- -- -- -- -- --

1 PSM/PSA 0.588 2.156 0.991 0.223 0.0007 0.142 100.00 92.59 

2 PSM/PSM 0.796 2.213 0.988 0.016 0.8128 0.119 100.00 100.00 

3 MFW/MFW 0.819 2.284 0.890 -0.007 0.9728 1.113 88.88 92.59 

4 MFW/PSM 0.584 2.093 0.926 0.228 0.1667 0.742 96.30 100.00 

5 MFW/PSA 0.377 2.026 0.932 0.435 0.0088 0.825 92.59 96.30 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Notes for Table 13: 

(1) All statistics in this table are based on 27 observations (1 data point for each of the years 1979 to 2005.) 

(2) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular year for 

the measure listed and the symbols in the denominator represent the value from the previous year. PSA = Petroleum Supply 

Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based on weekly data. 

(3) The values in the third, fourth, and sixth columns are year-to-year growth rates expressed as percentages. For example, 

for periods of the first 3 months, Method A has a mean of 0.960 percent and a standard deviation of 2.780 percent. 

(4) The symbol -- in the Method A row represents “Not Applicable”. 

(5) The correlations reported here are Pearson Product-Moment correlations of the year-to-year growth rates determined by

Methods 1 to 5 with the year-to-year growth rates for Method A. All correlations have a probability value (p-value) of 

<.0001. 

(6) The sixth column is mean differences in year-to-year growth rates. For example, the mean difference between Method A 

and Method 1 for first three months is 0.216 percent (that is, Method A’s mean is 0.216 percent higher.)

(7) The mean square error in the eighth column is equal to (Difference in Mean from Method A)2

+ (Standard Deviation of

the Difference from Method A from Table 15)2

. Its units are percent squared.

(8) The percentages reported in the ninth column are the percentage of times that each method gives an answer within 2% of

Method A. For example, for Method 3 for the periods of the first three months, 77.78 percent of the time it is within 2% of 

the Method A growth rate. 

(9) The last column gives the percentage of the time that each method gives a growth rate in the same direction as Method A. 

That is, it is the percentage of the time that each method and Method A both give positive growth rates or both give negative

growth rates. For example, for the periods of the first three months, 81.48 percent of the time Method 3 and Method A are 

either both positive or both negative. 

Significance Testing of Mean Differences in Growth Rates

The tests for deciding whether there were significant differences between the mean growth rate given by Method A and the

other five methods are given in Table 15. The differences in means and the p-values are also given in this table. Paired 

(matched) t-tests were used since the growth rates are indexed by year. It needs to be remembered that growth rate

differences, and not raw differences, are being tested here.

For the methods that use PSM as the numerator, Method 1’s (PSM/PSA) mean difference from Method A is significant while

Method 2’s (PSM/PSM) mean difference from Method A is not significant across for all time periods. However, the standard

deviations of the differences are smaller for Method 1 than for Method 2. To see the combined influence of the bias (mean 

difference from Method A) and the variability (variance of the differences), the mean square error, defined by (bias)2

+ 

(standard deviation of the differences)2 was computed. The mean square error for the periods of first three months is less for 

Method 1 than for Method 2. But, the mean square errors for the other time periods are greater for Method 1 than for Method

2. None of the differences in mean square errors between the two methods are meaningfully significant. Further, the mean

square errors for Methods 1 and 2 are much smaller than for Methods 3 to 5.

For the methods that use MFW as the numerator, Method 3 (MFW/MFW) has less bias than Method 4 (MFW/PSM) and 

Method 5 (MFW/PSA). But, Methods 4 and 5 have smaller standard deviations of the differences between them and Method

A than Method 3 does. Method 3 has a larger mean square error than Method 4 and Method 5. However, the mean square 

errors are close for Method 4 and Method 5 for all time periods. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 14. Descriptive Statistics for the Year-to-Year Growth Rates (as Percentages) for Finished

Motor Gasoline Supplied, 1979-2005 

Method Method in Standard 

Number Symbols Mean Deviation Median Minimum Maximum Range 

First 3 Months 

A PSA/PSA 0.960 2.780 1.845 -8.320 5.173 13.493 

1 PSM/PSA 0.744 2.675 1.303 -8.258 4.364 12.622 

2 PSM/PSM 0.946 2.752 1.387 -8.550 5.293 13.843 

3 MFW/MFW 1.004 3.140 1.318 -7.377 5.378 12.755 

4 MFW/PSM 0.892 2.689 1.396 -5.604 4.244 9.848 

5 MFW/PSA 0.692 2.650 0.412 -5.303 4.270 9.573 

First 6 Months

A PSA/PSA 0.847 2.255 1.451 -7.114 3.617 10.731 

1 PSM/PSA 0.569 2.136 1.192 -7.083 3.289 10.372 

2 PSM/PSM 0.837 2.229 1.366 -7.229 3.671 10.900 

3 MFW/MFW 0.889 2.482 1.199 -6.182 4.415 10.597 

4 MFW/PSM 0.447 2.240 0.816 -5.952 3.896 9.848 

5 MFW/PSA 0.179 2.128 0.394 -5.804 3.513 9.317 

First 9 Months

A PSA/PSA 0.821 2.164 1.365 -6.656 2.903 9.559 

1 PSM/PSA 0.569 2.081 1.107 -6.650 2.881 9.531 

2 PSM/PSM 0.806 2.143 1.363 -6.669 3.094 9.763 

3 MFW/MFW 0.815 2.346 1.228 -5.678 4.373 10.051 

4 MFW/PSM 0.583 2.025 0.898 -5.146 3.472 8.618 

5 MFW/PSA 0.346 1.954 0.665 -5.127 3.221 8.348 

Entire Year

A PSA/PSA 0.812 2.203 1.381 -6.242 2.932 9.174 

1 PSM/PSA 0.588 2.156 1.191 -6.286 2.736 9.022 

2 PSM/PSM 0.796 2.213 1.393 -6.237 3.018 9.255 

3 MFW/MFW 0.819 2.284 1.586 -5.152 3.927 9.079 

4 MFW/PSM 0.584 2.093 1.028 -4.879 3.168 8.047 

5 MFW/PSA 0.377 2.026 0.672 -4.928 2.970 7.898 

Notes: 

(1) All statistics in this table are based on 27 observations (one for each of the years 1979 to 2005.)

(2) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular quarter 

in a particular year for the measure listed and the symbols in the denominator represent the value from the same quarter in the

previous year. PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based

on weekly data. 

(3) The percentages reported in the third through eighth columns are for the year-to-year growth rates. For example, Method

A for the data for the first 3 months for the 27 years from 1979 to 2005 has a mean growth rate of 0.960 percent and a

standard deviation of 2.780 percent.

(4) Totals across components, where applicable, may not add due to independent rounding. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 15. Differences for Finished Motor Gasoline Supplied Between Method A and the Other Growth Rates,

1979-2005

Standard 

Largest Largest Deviation Mean 

Method Method in Under- Over- Mean of Square 

Number Symbols estimate estimate Difference Differences t p-value Error 

First 3 Months

1 PSM/PSA -1.259 0.623 0.216 0.413 -2.712 0.0117 0.217 

2 PSM/PSM -1.237 0.766 0.014 0.541 -0.136 0.8925 0.292 

3 MFW/MFW -4.793 2.350 -0.044 1.705 0.134 0.8947 2.910 

4 MFW/PSM -2.520 2.716 0.068 1.261 -0.280 0.7815 1.596 

5 MFW/PSA -2.675 3.017 0.268 1.297 -1.075 0.2921 1.754 

First 6 Months

1 PSM/PSA -0.162 1.538 0.277 0.411 3.514 0.0016 0.245 

2 PSM/PSM -1.344 0.992 0.009 0.471 0.103 0.9185 0.222 

3 MFW/MFW -3.104 2.678 -0.042 1.398 -0.155 0.8778 1.956 

4 MFW/PSM -1.258 3.008 0.400 1.112 1.869 0.0729 1.397 

5 MFW/PSA -1.309 3.256 0.668 1.049 3.308 0.0028 1.546 

First 9 Months

1 PSM/PSA -0.127 1.412 0.252 0.358 3.663 0.0011 0.192 

2 PSM/PSM -1.218 0.816 0.015 0.421 0.188 0.8523 0.177 

3 MFW/MFW -3.357 2.177 0.006 1.307 0.024 0.9807 1.707 

4 MFW/PSM -1.510 2.846 0.239 1.010 1.227 0.2308 1.078 

5 MFW/PSA -1.529 3.210 0.475 0.976 2.532 0.0177 1.178 

Entire Year

1 PSM/PSA -0.083 1.289 0.223 0.303 3.830 0.0007 0.142 

2 PSM/PSM -1.031 0.668 0.016 0.344 0.239 0.8128 0.119 

3 MFW/MFW -2.085 2.288 -0.007 1.055 -0.034 0.9728 1.113 

4 MFW/PSM -1.363 2.197 0.228 0.831 1.423 0.1667 0.742 

5 MFW/PSA -1.314 2.248 0.435 0.797 2.832 0.0088 0.825 

Notes: 

(1) All statistics in this table are based on 27 observations (one for each of the years 1979 to 2005.)

(2) To interpret the “Method in symbols” column, the symbols in the numerator represent the value from a particular month 

in a particular year for the measure listed and the symbols in the denominator represent the value from the same month in the 

previous year. PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based

on weekly data. 

(3) The units for Largest Underestimate, Largest Overestimate, Mean Difference, and Standard Deviation of the Difference 

are percent. The units for Mean Square Error are percent squared. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(4) The method that Methods 1 to 5 are being compared to is Method A: ⎛ Period (t) ⎞

A(Period ,t) = ⎜

PSA − 1⎟ *100% , where ⎜ ⎟ ⎝ Period PSA (t − 1) ⎠

Period = first 3 months, first 6 months, first 9 months, or the entire year. The percentages reported are differences in year-toyear growth rates. For example, the mean difference for year-to-year growth rate for the first 3 months for the 27 years of

1979 to 2005 between Method A and Method 1 (PSM/PSA) is 0.216 percent (that is, Method A’s mean is 0.216 percent 

higher.) 

(5) Mean Square Error = (bias)2

+ (standard deviation)2. Here the bias = Mean Difference (the fifth column) and the standard

deviation = Standard Deviation of the Differences (the sixth column.) 

Percentage of the Time Within 1% and 2% of the Method A Growth Rate 

Table 16 reports the percentages of the time that each method is within 1% and 2% of Method A. Methods 1 and 2 are within

1% of Method A almost all of the time (lowest percentage is 88.89 percent, which is 24 out of 27 years) and always within

2% of Method A. While the percentages differ slightly between the time periods for those methods that use MFW as the

numerator, the patterns are the same: Method 3 (MFW/MFW) has a lower percentage of years that it is within 1% and within

2% of Method A than Method 4 (MFW/PSM) and Method 5 (MFW/PSA) do.

Occurrence of Reported Correct and Wrong Directions of Growth Rates 

As discussed in Study 2, sometimes for year-to-year growth rates it occurs that some methods yield a positive growth rate 

and other methods yield a negative growth rate for the same time period. As seen in Tables 13 and 17, Method 1 (PSM/PSA) 

gives the same direction as Method A between 92.59 percent (25/27) and 96.30 percent (26/27) of the time. Method 2 

(PSM/PSM) gives the same direction as Method A between 96.30 percent and 100.00 percent of the time. Methods 3

(MFW/MFW) is not as good on this criterion. It gives the same direction as Method A between 81.48 percent and 96.30 

percent of the time. Method 4 (MFW/PSM) and Method 5 (MFW/PSA) are in the middle. They give the same direction as

Method A between 85.19 percent and 100.00 percent of the time. These results must be interpreted carefully since there are

only 27 observations for each time period

Main Results from Study 3 

• Method 1 (PSM/PSA) and Method 2 (PSM/PSM) are preferable to all of the methods that use MFW as their numerator 

(Methods 3 to 5.) They have higher correlations with Method A; their standard deviations are closer to Method A’s than

those for Methods 3 to 5 are; their mean square errors are smaller; they are within 1% and 2% of Method A more often; and

they give the correct direction slightly more often.

• It is hard to decide between Method 1 and Method 2 as to which is better. There is a slight preference for Method 2 since 

it has a lower mean square error for all time periods, except for the time period of the first 3 months. 

• Method 4 (MFW/PSM) and Method 5 (MFW/PSA) are preferable to Method 3 (MFW/MFW). They have higher 

correlations with Method A; their mean square errors are smaller; they are within 1% and 2% of Method A more often; and 

they give the correct direction slightly more often.

• It is hard to decide between Method 4 and Method 5 here. Method 5 has slightly lower correlations with Method A than

Method 4 does. Also, Method 5 has lower p-values than Method 4 for the differences in means between it and Method A. 

Otherwise, there are no meaningful differences between Methods 4 and 5. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 16. Percentage of Time Each Year-to-Year Growth Rate Method is Within 1% and 2% of 

Method A for Finished Motor Gasoline Supplied, 1979-2005 

Percent 

Percent Percent Percent Percent Percent Percent of the

of the of the of the of the of the of the Time 

Time Time Time Time Time Time Not 

Method Method in Within Within Within Within Within Within Within 

Number Symbols -1% +1% -2% +2% 1% 2% 2%

First 3 Months

1 PSM/PSA 70.37 25.93 74.07 25.93 96.30 100.00 0.00 

2 PSM/PSM 29.63 59.26 40.74 59.26 88.89 100.00 0.00 

3 MFW/MFW 11.11 22.22 29.63 48.15 33.33 77.78 22.22 

4 MFW/PSM 18.52 33.33 44.44 48.15 51.85 92.59 7.41 

5 MFW/PSA 33.33 22.22 55.56 33.33 55.56 88.89 11.11 

First 6 Months

1 PSM/PSA 29.63 59.26 29.63 70.37 88.89 100.00 0.00 

2 PSM/PSM 55.56 40.74 59.26 40.74 96.30 100.00 0.00 

3 MFW/MFW 33.33 25.93 48.15 33.33 59.26 81.48 18.52 

4 MFW/PSM 33.33 33.33 40.74 48.15 66.67 88.89 11.11 

5 MFW/PSA 22.22 51.85 25.93 62.96 74.07 88.89 11.11 

First 9 Months

1 PSM/PSA 29.63 66.67 29.63 70.37 96.30 100.00 0.00 

2 PSM/PSM 40.74 55.56 44.44 55.56 96.30 100.00 0.00 

3 MFW/MFW 37.04 22.22 44.44 40.74 59.26 85.18 14.82 

4 MFW/PSM 44.44 29.63 48.15 44.44 74.07 92.59 7.41 

5 MFW/PSA 25.93 48.15 29.63 62.96 74.07 92.59 7.41 

Entire Year

1 PSM/PSA 18.52 77.78 18.52 81.48 96.30 100.00 0.00 

2 PSM/PSM 48.15 48.15 51.85 48.15 96.30 100.00 0.00 

3 MFW/MFW 40.74 29.63 44.44 44.44 70.37 88.88 11.12 

4 MFW/PSM 40.74 40.74 44.44 51.85 81.48 96.30 3.70 

5 MFW/PSA 18.52 62.96 22.22 70.37 81.48 92.59 7.41 

Notes: 

(1) All statistics in this table are based on 27 observations (one for each of the years 1979 to 2005.)

(2) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular month 

in a particular year for the measure listed and the symbols in the denominator represent the value from the same month in the 

previous year. PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based

on weekly data. 
(3) The percentages reported are percentage of times that each method gives an answer within 1% or 2% (depending on the

column) of A Method ⎛ Period (t) ⎞= ⎜

PSA − 1⎟ *100% , where Period = first 3 months, first 6 months, first 9 months, or the ⎜

⎝ Period PSA (t − 1) ⎟

⎠

entire year. For example, the first two entries for the Method 3 row for first 3 months mean that 11.11 percent of the

MFW/MFW ratios are below the Method A ratios, but are still within 1% of them and that 22.22 percent of the MFW/MFW

ratios are above the Method A ratios, but are still within 1% of them. 

(4) Totals across components, where applicable, may not add to 100 percent due to independent rounding. 

Table 17. Occurrence of Reported Correct and Wrong Directions of Growth Rates for Finished Motor Gasoline 

 Supplied, 1979 to 2005

 

 

 

Percent of the

Time that 

Method A and 

Percent of Percent of the Percent of the the Other

Percent of Time Both Time Method A Time Method A Method are in 

Method 

Number 

Method in 

Symbols 

Time Both 

Measures Up 

Measures 

Down

is Up and the 

Other is Down 

is Down and 

the Other is Up 

the Same

Direction 

First 3 Months

1 PSM/PSA 66.67 29.63 0.00 3.70 96.30 

2 PSM/PSM 66.67 29.63 0.00 3.70 96.30 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 MFW/MFW 59.26 22.22 7.41 11.11 81.48 

4 MFW/PSM 62.96 25.93 3.70 7.41 88.89 

5 MFW/PSA 62.96 33.33 3.70 0.00 96.30 

First 6 Months

1 PSM/PSA 62.96 33.33 3.70 0.00 96.30 

2 PSM/PSM 66.67 29.63 0.00 3.70 96.30 

3 MFW/MFW 59.26 22.22 7.41 11.11 81.48 

4 MFW/PSM 62.96 29.63 3.70 3.70 92.59 

5 MFW/PSA 55.56 29.63 11.11 3.70 85.19 

First 9 Months

1 PSM/PSA 66.67 25.93 7.41 0.00 92.59 

2 PSM/PSM 74.07 25.93 0.00 0.00 100.00

3 MFW/MFW 74.07 22.22 0.00 3.70 96.30 

4 MFW/PSM 70.37 25.93 3.70 0.00 96.30 

5 MFW/PSA 70.37 25.93 3.70 0.00 96.30 

Entire Year

1 PSM/PSA 66.67 25.93 7.41 0.00 92.59 

2 PSM/PSM 74.07 25.93 0.00 0.00 100.00

3 MFW/MFW 70.37 22.22 3.70 3.70 92.59 

4 MFW/PSM 74.07 25.93 0.00 0.00 100.00

5 MFW/PSA 70.37 25.93 3.70 0.00 96.30 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Notes for Table 17: 

(1) All statistics in this table are based on 27 observations (one for each of the years 1979 to 2005.)

(2) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular month 

in a particular year for the measure listed and the symbols in the denominator represent the value from the same month in the 

previous year. PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based

on weekly data. 

(3) The percentages reported are percentage of times that each method gives an answer within 1% or 2% (depending on the

⎛ Period (t) ⎞

column) of Method A of A(Period ,t) =

⎜

⎜

PSA − 1

⎟

⎟ *100% , where Period = first 3 months, first 6 months, first 9

Period (t − 1) ⎝ PSA ⎠

months, or the entire year. For example, the first two entries for the Method 3 row in the First 3 Months portion of the table

mean that 59.26 percent of the MFW/MFW ratios and Method A ratios are both positive (that is, indicate upward growth

rates) and that 22.22 percent of the MFW/MFW ratios and the Method A ratios are both negative (that is, indicate downward

growth rates.) 

(4) Totals across components, where applicable, may not add due to independent rounding. 

STUDY 4 

Comparing Method 4 and Method 5 Directly to Method 2

In Study 1, Study 2 and Study 3 it was concluded that using Method 2 (PSM/PSM) was the best approximation to Method A 

(PSA/PSA) and that no conclusion could be made as to whether Method 4 or Method 5 was the better approximation. Since

it is expected that in the future, based on the results of these studies, Method 2 will be used more often by EIA and others for 

product supplied growth rates, it was decided to compare Method 4 and Method 5 directly to Method 2. By comparing 

Method 4 and Method 5 directly to Method 2, it was hoped that one of these two methods would be shown to be superior.

One way of thinking of what is going on is that the previous work compared the other methods to a “gold standard” of

Method A. This study compares Methods 4 and 5 to a “silver standard” of Method 2.

In order to compare Methods 4 and 5 to Method 2, tables similar to those from the previous studies were made. Tables 18

and 19 give summary statistics and paired t-test results for the differences between Methods 4 and 5 and Method 2. Tables

20 and 21 give the percentage of times that the growth rates computed by Methods 4 and 5 are within 1% and 2% of Method

2’s growth rates. Table 22 gives the percent agreement in terms of direction (either positive or negative) of the Method 4 and

5 growth rates with those of Method 2. 

From inspection of these tables it is now clear that Method 4 is preferable to Method 5. Method 4 has smaller mean square 

errors than Method 5 for all comparison periods. In terms of the percentage of times that each method is within 2% of

Method 2, 75 percent of the time (6 out of 8 comparison periods) Method 4 has a higher percentage than Method 5 and 12.5

percent (1 out of 8 comparison periods) of the time is tied with Method 5. For 87.04 percent of the quarters from 1978 to

2005, Method 4 gives a growth rate in the same direction as Method 2 while Method 5 gives a growth rate in the same

direction only 80.56 percent of the time. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 18: Differences Between Method 2 versus Method 4 and Method 5 for Quarterly Growth Rates 

for Finished Motor Gasoline Supplied, 1979-2005 

Standard 

Method 

Number 

Method in 

Symbols 

Largest 

Underestimate 

Largest 

Overestimate Mean 

Difference 

Deviation 

of

Differences t p-value

Mean 

Square 

Error 

All 108 Quarters 

4 MFW/PSM -5.472 2.946 0.209 1.234 -1.762 0.0810 1.567 

5 MFW/PSA -5.712 3.247 0.417 1.312 -3.303 0.0013 1.894 

First Quarters (n =27) 

4 MFW/PSM -1.779 2.946 0.054 1.212 -0.231 0.8193 1.472 

5 MFW/PSA -2.682 3.247 0.254 1.360 -0.971 0.3403 1.914 

Second Quarters (n = 27) 

4 MFW/PSM -5.472 1.553 0.709 1.472 -2.503 0.0189 2.668 

5 MFW/PSA -5.712 0.778 1.040 1.489 -3.630 0.0012 3.300 

Third Quarters (n = 27) 

4 MFW/PSM -3.346 2.445 -0.099 1.161 0.444 0.6605 1.358 

5 MFW/PSA -3.638 2.431 0.080 1.254 -0.331 0.7431 1.579 

Fourth Quarters (n = 27) 

4 MFW/PSM -1.965 1.850 0.174 0.951 -0.949 0.3515 0.934 

5 MFW/PSA -2.437 1.937 0.293 0.936 -1.625 0.1162 0.962 

Notes: 

(1) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular quarter 

in a particular year for the measure listed and the symbols in the denominator represent the value from the same quarter in the

previous year. PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based

on weekly data. 

(2) The units for Largest Underestimate, Largest Overestimate, Mean Difference, and Standard Deviation of the Difference 

are percent. The units for Mean Square Error are percent squared. 

(3) The method that Methods 4 and 5 are being compared to is Method 2 by ⎛defined Quarter PSM (t) ⎞

⎜ − 1⎟ *100% . The ⎜

⎝ Quarter PSM (t − 1) ⎟

⎠

percentages reported in the fifth column are differences in year-to-year growth rates. For example, for First Quarters, Method

2’s mean is .054 percent higher than Method 4’s mean. 

(4) Mean Square Error = (bias)2

+ (standard deviation)2

. Here the bias = Mean Difference (the fifth column) and the standard

deviation = Standard Deviation of the Differences (the sixth column.) 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 19: Differences Between Method 2 versus Method 4 and Method 5 for Growth Rates for 

3-, 6-, 9-, and 12-month Periods for Finished Motor Gasoline Supplied, 1979-2005 

Standard 

Method 

Number 

Method in 

Symbols 

Largest 

Underestimate 

Largest 

Overestimate Mean 

Difference 

Deviation 

of

Differences t p-value

Mean 

Square 

Error 

First 3 Months

4 MFW/PSM -1.779 2.946 0.054 1.212 -0.231 0.8193 1.472 

5 MFW/PSA -2.682 3.247 0.254 1.360 -0.971 0.3403 1.914 

First 6 Months

4 MFW/PSM -3.122 1.487 0.391 1.121 -1.812 0.0816 1.409 

5 MFW/PSA -3.369 1.424 0.658 1.158 -2.954 0.0066 1.775 

First 9 Months

4 MFW/PSM -2.793 1.522 0.223 0.990 -1.172 0.2519 1.031 

5 MFW/PSA -3.157 1.541 0.460 1.044 -2.290 0.0304 1.302 

Entire Year

4 MFW/PSM -2.171 1.358 0.212 0.804 -1.368 0.1829 0.691 

5 MFW/PSA -2.241 1.309 0.419 0.844 -2.579 0.0159 0.887 

Notes: 

(1) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular time

period in a particular year for the measure listed and the symbols in the denominator represent the value from the same time

period in the previous year. PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly

estimates based on weekly data. 

(2) The units for Largest Underestimate, Largest Overestimate, Mean Difference, and Standard Deviation of the Difference 

are percent. The units for Mean Square Error are percent squared. 

⎛ Period (t) ⎞

(3) The method that Methods 4 and 5 are being compared to is Method 2 defined by ⎜

⎜

PSM − 1

⎟

⎟ *100% . The 

⎝ Period PSM (t − 1) ⎠

percentages reported in the fifth column are differences in year-to-year growth rates. For example, for periods of the First 3

Months, Method 2’s mean is .054 percent higher than Method 4’s mean. 

(4) Mean Square Error = (bias)2

+ (standard deviation)2. Here the bias = Mean Difference (the fifth column) and the standard

deviation = Standard Deviation of the Differences (the sixth column.) 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 20. Percentage of Time Methods 4 and 5 are Within 1% and 2% of Method 2 Over All

Quarters for Finished Motor Gasoline Supplied, 1979-2005

Percent 

Percent Percent Percent Percent Percent Percent of the

of the of the of the of the of the of the Time 

Time Time Time Time Time Time Not 

Method Method in Within Within Within Within Within Within Within 

Number Symbols -1% +1% -2% +2% 1% 2% 2%

All 108 Quarters 

4 MFW/PSM 37.04 28.70 52.78 37.96 65.74 90.74 9.26 

5 MFW/PSA 31.48 25.93 52.78 36.11 57.41 88.89 11.11 

First Quarters (n =27) 

4 MFW/PSM 40.74 18.52 59.26 33.33 59.26 92.59 7.41 

5 MFW/PSA 37.04 7.41 66.67 25.93 44.44 92.59 7.41 

Second Quarters (n = 27) 

4 MFW/PSM 37.04 29.63 48.15 33.33 66.67 81.48 18.52 

5 MFW/PSA 25.93 29.63 44.44 29.63 55.56 74.07 25.93 

Third Quarters (n = 27) 

4 MFW/PSM 48.15 22.22 51.85 37.04 70.37 88.89 11.11 

5 MFW/PSA 29.63 33.33 40.74 51.85 62.96 92.59 7.41 

Fourth Quarters (n = 27) 

4 MFW/PSM 22.22 44.44 51.85 48.15 66.67 100.00 0.00 

5 MFW/PSA 33.33 33.33 59.26 37.04 66.67 96.30 3.70 

Notes: 

(1) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular quarter 

in a particular year for the measure listed and the symbols in the denominator represent the value from the same quarter in the

previous year. PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based

on weekly data. 

(2) The percentages reported are percentage of times that each method gives an answer within 1% or 2% (depending on the

⎛ Quarter PSM (t) ⎞

column) of Method 2 defined by − . 1 *100% ⎜

⎜

⎟

⎟

Quarter (t − 1) ⎝ PSM ⎠

(3) Totals across components, where applicable, may not add to 100 percent due to independent rounding. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 21. Percentage of Time Methods 4 and 5 are Within 1% and 2% of Method 2 Over 

3-, 6-, 9- and 12-month Periods for Finished Motor Gasoline Supplied, 1979-2005 

Percent 

Percent Percent Percent Percent Percent Percent of the

of the of the of the of the of the of the Time 

Time Time Time Time Time Time Not 

Method Method in Within Within Within Within Within Within Within 

Number Symbols -1% +1% -2% +2% 1% 2% 2%

First 3 Months

4 MFW/PSM 40.74 18.52 59.26 33.33 59.26 92.59 7.41 

5 MFW/PSA 37.04 7.41 66.67 25.93 44.44 92.59 7.41 

First 6 Months

4 MFW/PSM 33.33 29.63 48.15 40.74 62.96 88.89 11.11 

5 MFW/PSA 37.04 22.22 55.56 29.63 59.26 85.19 14.81 

First 9 Months

4 MFW/PSM 37.04 44.44 44.44 48.15 81.48 92.59 7.41 

5 MFW/PSA 44.44 33.33 51.85 37.04 77.78 88.89 11.11 

Entire Year

4 MFW/PSM 44.44 37.04 55.56 40.74 81.48 96.30 3.70 

5 MFW/PSA 48.15 29.63 59.26 33.33 77.78 92.59 7.41 

Notes: 

(1) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular time

period in a particular year for the measure listed and the symbols in the denominator represent the value from the same time

period in the previous year. PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly

estimates based on weekly data. 

(2) The percentages reported are percentage of times that each method gives an answer within 1% or 2% (depending on the

⎛ Period (t) ⎞

column) of Method 2 defined by PSM − 1 *100% .

⎜

⎜

⎟

⎟

⎝ Period PSM (t − 1) ⎠

(3) Totals across components, where applicable, may not add to 100 percent due to independent rounding. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 22. Occurrence of Reported Same and Different Directions for Growth Rates using Method 2, Method 4, and 

Method 5 over All Quarters for Finished Motor Gasoline Supplied, 1979 to 2005 

Percent of the

Time that 

Percent of the Method 2 and

Percent of the Time Method 2 Percent of the Percent of the the Other

Time Method 2 and the Other Time Method 2 Time Method 2 Method are in 

Method Method in and the Other Method are is Up and the is Down and the the Same

Number Symbols Method are Up Down Other is Down Other is Up Direction 

4 MFW/PSM 66.67 20.37 9.26 3.70 87.04 

5 MFW/PSA 60.19 20.37 15.74 3.70 80.56 

Notes: 

(1) All statistics in this table are based on 108 observations (four quarters for each of the years 1979 to 2005.) 

(2) To interpret the “Method in Symbols” column, the symbols in the numerator represent the value from a particular month 

in a particular year for the measure listed and the symbols in the denominator represent the value from the same month in the 

previous year. PSA = Petroleum Supply Annual; PSM = Petroleum Supply Monthly; and MFW = Monthly estimates based

on weekly data. 

(3) The percentages reported are percentage of times that each method gives an answer within 1% or 2% (depending on the

⎛ Quarter (t) ⎞

column) of Method 2 defined by: PSM − 1 *100% .

⎜

⎜

⎟

⎟

Quarter (t − 1) ⎝ PSM ⎠

(4) Totals across components, where applicable, may not add due to independent rounding. 

Overall Main Results (The Big Picture) 

• Method 2 using PSM for a given time period in one year divided by the PSM value for the same period in the previous

year is preferable to Method 1 that uses PSM divided by PSA. 

• Method 3 (MFW/MFW) is inferior to Method 4 (MFW/PSM) and Method 5 (MFW/PSA). 

• Method 4 is preferable to Method 5, but it is a close call. This preference is based on the results of Study 4. 

• The above three results when combined have a nice succinct conclusion when Method A is not possible to use. That is, 

when Method A cannot be used, then always use PSM measurements from the previous year as the denominator when 

computing growth rates for any time period. This conclusion is for both PSM and MFW as the numerator.

Limitations 

• The analyses done were only for finished motor gasoline supplied and distillate fuel oil supplied, although only the ones 

for finished motor gasoline were discussed in this paper. Other product supply measures were not analyzed. 

• The analyses of the growth rates from Blumberg (2007) with ratios of one-month time periods (summarized here as Study

1) only used data from the period of 1995 to 2004. These analyses need to be redone using the 27 years that were used in this

paper. 
 

 

 

 

 

 

 

 

 

 

 

 

 

Directions for Future Research 

• Similar analyses should be carried out for the equivalent data from the Petroleum Marketing surveys and for other

Petroleum Supply measures. 

• An analysis of the different methods of estimating month-to-month growth rates between two consecutive months should

be carried out using similar analyses to those used here. 

• The rate of change of the growth rates (that is, the counterpart of the second derivative (concavity) when looking at

continuous functions) may be worth investigating.

Acknowledgements 

The idea for Study 1 came from John Cook, Director of the Petroleum Division. The ideas for Study 2 and Study 3 came 

from Joanne Shore. Both of them, as well as Carol French and Douglas MacIntyre, helped in the design of the methodology

of these studies. Tammy Heppner and Bin Zhang provided the data needed to carry out this research. 

References 

Blumberg, Carol J. “Comparison of Different Methods of Computing Yearly Growth Rates for Petroleum Supply, 1995-

2004.” Unpublished paper prepared for the Spring 2007 meeting of the American Statistical Association Committee on 

Energy Statistics, April 5 to 6, 2007 (Washington DC, January 30, 2007.) Located at 

http://www.eia.doe.gov/smg/asa_meeting_2007/spring/files/comparemethods.pdf. 

Kirkendall, Nancy and Donald Martin. “Comparison of Choletta-Chhab and MFW Methods.” Unpublished paper prepared

for the Spring 1993 meeting of the American Statistical Association Committee on Energy Statistics (Washington DC, 

February 23, 1993.)