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i

NASA/TP—2014–218192

Comparison of the Variations of Sunspot

Number, Number of Sunspot Groups,

and Sunspot Area, 1875–2013

Robert M. Wilson

Marshall Space Flight Center, Huntsville, Alabama

April 2014

National Aeronautics and

Space Administration

Marshall Space Flight Center • Huntsville, Alabama 35812
ii

Available from:

NASA STI Information Desk

Mail Stop 148

NASA Langley Research Center

Hampton, VA 23681–2199, USA

757–864–9658

This report is also available in electronic form at

<http://www.sti.nasa.gov>
iii

TABLE OF CONTENTS

1. INTRODUCTION ............................................................................................................. 1

2. RESULTS ........................................................................................................................... 3

3. DISCUSSION AND SUMMARY ...................................................................................... 19

REFERENCES ....................................................................................................................... 21
iv

LIST OF FIGURES

 1. Annual variation of (a) SSN, (b) G, and (c) SSA for the interval 1875–2013 ............ 4

 2. SC24 values of (a) SSN, (b) G, (c) SSA, and (d) NSD in relation to the mean 

of cycles 12–23 for elapsed time t=–3 to 10 years from SSNmin .............................. 6

 3. Annual variation of the ratios (a) SSN/G, (b) SSA/G, and (c) SSA/SSWN 

for the interval 1875–2013 ........................................................................................ 7

 4. SC24 values of the ratios (a) SSN/G, (b) SSA/G, and (c) SSA/SSN in relation 

to the mean of cycles 12–23 for elapsed time t=–3 to 10 years from SSNmin .......... 9

 5. SC-mean values of (a) SSN, (b) G, (c) SSA, (d) SSN/G, (e) SSA/G, 

and (f) SSA/SSN for SC12–SC23 ............................................................................. 10

 6. SC24 values of (a) SSN, (b) G, (c) SSA, (d) SSN/G, (e) SSA/G, 

and (f) SSA/SSN in relation to SC-mean values (SC12–SC23) for elapsed 

time t=0–10 years .................................................................................................... 11

 7. Scatter plot of (a) SSN versus G and (b) SSA versus G for the interval 

1875–2013 ................................................................................................................ 13

 8. Scatter plot of SSN versus SSA for the interval 1875–2013 ...................................... 14

 9. Scatter plot of SSN (observed) versus SSN (predicted, G and SSA) 

for the interval 1875–2013 ........................................................................................ 15

10. Annual variation of the difference (observed – predicted) for the interval 

1875–2013 ................................................................................................................ 16
v

LIST OF TABLES

1. Minimum and maximum parametric values and epochs of occurrence, 

SC12–SC24 ............................................................................................................... 5

2. Results of statistical testing of the parametric means based on the first 6 years 

of SC14 and SC24 using the t statistic for independent samples ................................ 12

3. Comparison of observed and predicted SSN, where predicted SSN 

is the bivariate fit of SSN based on G and SSA ......................................................... 17
vi

LIST OF ABBREVIATIONS, ACRONYMS, AND DESIGNATORS

A area (relation between sunspot area and sunspot number)

ASC ascent duration in years

G number of sunspot groups

<G> mean number of sunspot groups

Gmin minimum number of sunspot groups

Gmax maximum number of sunspot groups

M cycle maximum

m cycle minimum

N number of individual spots within spot groups

NOAA National Oceanic and Atmospheric Administration

NSD number of spotless days

<NSD> mean number of spotless days

NSDmax maximum number of spotless days

PER period or cycle length (from SSNmin to next cycle SSNmin)

R relative sunspot number

RGO Royal Greenwich Observatory

SC sunspot cycle

SIDC Solar Influences and Data Analysis Center

SOON Solar Optical Observing Network

SSA sunspot area
vii

LIST OF ABBREVIATIONS, ACRONYMS, AND DESIGNATORS (Continued)

<SSA> mean sunspot area

SSA/G ratio of SSN to G

<SSA/G> mean ratio SSN to G

SSAmax maximum sunspot area

SSAmin minimum sunspot area

SSA/SSN ratio of SSA to SSN

<SSA/SSN> mean ratio of SSA to SSN

SSN sunspot number

<SSN> mean sunspot number

SSN/G ratio of SSN to G

<SSN/G> mean ratio of SSN to G

SSNmax maximum sunspot number

SSNmin minimum sunspot number

USAF United States Air Force
viii

NOMENCLATURE

cl confidence level

k personal reduction coefficient

Ry12 coefficient of the sample multiple correlation

r coefficient of correlation

r 2 coefficient of determination

sd standard deviation

Sy12 coefficient of the sample multiple correlation error of estimate

se standard error of estimate

t elapsed time from SSNmin; the t statistic for independent samples

x independent variable

x1 first independent variable in the bivariate fit

x2 second independent variable in the bivariate fit

y dependent variable

z normal deviate for the sample
1

TECHNICAL PUBLICATION

COMPARISON OF THE VARIATIONS OF SUNSPOT NUMBER, NUMBER OF SUNSPOT 

GROUPS, AND SUNSPOT AREA, 1875–2013

1. INTRODUCTION

Long ago1–3 Samuel Heinrich Schwabe, a German apothecary and amateur astronomer, 

demonstrated the existence of the sunspot cycle, a 10- to 12-year variation in the number of sunspots 

with time (i.e., the elapsed time from cycle minimum to next cycle minimum). He accomplished this 

by recording the number of ‘days when no spots were observed’ and the number of ‘clusters of spots’ 

that he observed daily over the course of the year from Dessau, Germany, during the interval of 

1826–1868, averaging about 290 observing days per year. He reasoned that the peak annual number 

of spotless days represented a minimum of sunspot activity (occurring concurrently with the minimum annual number of clusters of spots), while the peak annual number of clusters of spots represented a maximum of sunspot activity. In 1857 the Royal Astronomical Society awarded Schwabe its 

gold medal and in 1868 Schwabe was elected a member of the Royal Society of London.4

Unfortunately, the discovery of the sunspot cycle went largely unnoticed for several years. In 

fact, it was not until about 1851 when Friedrich Wilhelm Heinrich Alexander von Humboldt mentioned it in his Kosmos (vol. 3), a treatise on science and nature, that the existence of the sunspot 

cycle was able to finally draw more widespread attention.5

It should be noted, however, that about 1847 Johan Rudolf Wolf, a professional astronomer 

and Director of the Berne Observatory, had become aware of Schwabe’s work on sunspots and in 

1849 he began his own systematic recording of sunspot counts based upon the use of his ‘relative 

sunspot number,’ an index he had introduced in 1848.6–8 Wolf determined his relative sunspot number 

from the observed number of sunspot groups (G, essentially the same as Schwabe’s clusters of spots) 

and the total number of individual spots (N) within all groups, computing it as R=10G+N, where 

R represents the relative sunspot number. Whereas Schwabe initially published his daily findings as 

annual values once per year, Wolf instead published his daily findings as monthly values twice per year 

(between 1849 and 1855). He also recorded the quality of sky conditions (regarding cloudiness) and 

which of the two telescopes (small or large) he had used in the determination of his relative sunspot 

number. As to why the number 10 is used instead of some other number,8 Wolf said, “I chose the 

number 10, on the one hand, because it seemed to work well for a large number of instances which I 

studied for this purpose, and, on the other hand, because it was simpler to deal with than any other 

number close to it.” (Wolf apparently had preferred to measure the area of sunspots7 for describing 

the sunspot cycle rather than determining his relative sunspot number, but because of his lack of the 

necessary instrumentation needed to perform such detailed measurements, his relative sunspot number, by default, became the index of choice regarding the description of the sunspot cycle.)
2

Today, the relative sunspot number remains the main descriptor of the sunspot cycle, primarily because of its continued use for the past 165 years, although one cannot deny the arbitrariness 

of the method of using G and N (rather than sunspot area). However, as noted by Clette et al.,9 the 

historical record of sunspot number actually is a blend of various sunspot indices of different intrinsic nature and accuracy, with systematic observations extending only from 1848 to the present: Wolf 

sunspot numbers spanning the interval 1848–1882, Zürich sunspot numbers spanning the interval 

1882–1980, and Solar Influences Data Analysis Center (SIDC) sunspot numbers (also referred to as 

International sunspot number) spanning the interval 1981–present. Consequently, relative sunspot 

number is now calculated as R = k(10G+N), where k is a personal reduction coefficient calculated 

for each individual observer to bring observations into the semblance of agreement. For the Wolf 

timespan, k=1; for the Zürich timespan, k=0.6 (changed to account for the inclusion of smaller 

spots and weighted according to size and structure of the umbra); and for the SIDC timespan, k is 

variable, averaging about 0.9. Other sunspot indices10 include the Boulder sunspot number, having 

k=1, and the American sunspot number, having a variable k that averages about 0.7. The Boulder 

number typically measures about 25% larger than the International number. More recently, Hoyt 

and Schatten have introduced the Group sunspot number, which essentially mimics sunspot numbers 

from 1882 onwards, but which can depart by as much as 25% from Wolf’s reconstructed sunspot 

numbers for the interval prior to 1882 (i.e., the Wolf regime and historical reconstructed interval, 

back to 1610).11–15

The measurement of sunspot areas did not become routinely available until 1874 and later, due 

to the efforts of the Royal Greenwich Observatory (RGO).6,16,17 Beginning in May 1874, the RGO 

began cataloging daily sunspot positions and areas, based on systematic observations at Greenwich, 

England, Cape Town, South Africa, and Kodaikanal, India. The data were obtained directly from 

photographs by means of reticules divided into small squares and measured relative to the Sun’s center. Unfortunately, the RGO dataset ended in 1976. (From the RGO data, one determines both the 

number of sunspot groups and the corrected sunspot area, corrected for foreshortening.)

Continuation of the sunspot area dataset beyond 1976 is accomplished today through the use 

of the worldwide United States Air Force (USAF)/National Oceanic and Atmospheric Administration 

(NOAA) Solar Optical Observing Network (SOON). Sunspot areas from the USAF/NOAA/SOON system are visual determinations rather than photographic determinations. Studies16,18 have indicated that 

the visual determinations of sunspot area underestimate the area as determined from photographs by 

about 40%. Hence, one must multiply the observed visually determined sunspot area by the factor 1.4.

This study examines the variations of sunspot number (SSN), the number of sunspot groups 

(G), and the corrected sunspot area (SSA) for the interval 1875–2013, with the primary intention 

being the determination of the formulation for an expression that relates sunspot number to both 

the combined number of sunspot groups and sunspot area. This study also examines (1) the ratios 

SSN/G, SSA/G, and SSA/SSN for the interval 1875–2013; (2) sunspot cycle (SC) 24’s SSN, G, SSA, 

number of spotless days (NSD), <SSN/G>, <SSA/G>, and <SSA/SSN> in comparison with the 

means and standard deviations (sd) of these parameters for elapsed times (t) –3 to 10 years from 

SSN minimum amplitude occurrence; (3) the variation of cyclic averages of <SSN>, <G>, <SSA>, 

<SSN/G>, <SSA/G>, and <SSA/SSN> for SC12–SC23; and (4) the comparison of parametric 

means based on the first 6 years of SC14 and SC24.
3

2. RESULTS

Figure 1 displays the annual values of (a) SSN, (b) G, and (c) SSA (in millionths of a solar 

hemisphere) for the interval 1875–2013. Figure 1(a) also delineates the SC number, spanning the last 

3 years of SC11 through the first 6 years of SC24, the current ongoing SC. Sunspot cycles are reckoned from cycle minimum to next cycle minimum (i.e., from SSN minimum amplitude of a cycle to 

the next cycle’s SSN minimum amplitude, or using NSD, from NSD maximum amplitude of a cycle 

to the next cycle’s NSD maximum amplitude), which corresponds to its period (PER) or cycle length 

that has always been 10–12 years during the modern era of sunspot observations.19–25 For the modern era, sunspot cycles are found to have ascent (ASC) lengths (i.e., the elapsed time from a cycle’s 

SSN minimum amplitude to its SSN maximum amplitude) of 3–5 years.26 On average, during the 

interval 1875–2013, SSN measures 54.8 (±45), G measures 4.74 (±3.69), and SSA measures 838.8 

(±745) millionths of a solar hemisphere, where the values within the parentheses are the parametric 

standard deviations (sd). (SSN is available online at http://sidc.oma.be/index.php3 and SSA and G 

are available online at http://solarscience.msfc.nasa.gov/greenwch.shtml.) 

Table 1 gives the minimum and maximum parametric values and the dates of their occurrence for 

each of SC12–SC24, as well as each cycle’s ASC and PER (based on SSN). Plainly, the weakest cycle to 

date (i.e., smallest SSNmax) is SC14, having SSNmax=63.5, and the strongest cycle to date (i.e., largest 

SSNmax) is SC19, having SSNmax=190.2. In terms of SSNmax, SC24 appears to be slightly stronger 

than either SC12 (SSNmax=63.7) or 14, based on its preliminary SSN value for 2013 (64.9), inferring 

an ASC=5 years. SC24 is expected to persist through at least 2017 and possibly through 2019. The 

reader should notice that, while all Gmin occurrence dates are concurrent with the occurrence dates of 

SSNmin, only 8 of 12 cycles have had concurrent dates for SSNmax and Gmax, the exceptions include 

SC12, SC14, SC20, and SC22. Thus, there is the possibility that Gmax for SC24 has not yet been seen 

and might occur 1–2 years after the occurrence of SSNmax (presumed to be 2013). For SSAmin, only 

SC21 has an occurrence date different from the occurrence of SSNmin. For SSAmax, all cycles except 

SC20, SC21, and SC23 have occurrence dates concurrent with the occurrence dates of SSNmax. For 

NSDmax, all dates of occurrence are concurrent with the occurrence dates of SSNmin. In terms of 

SSAmax, SC24 is the smallest cycle to date, having SSAmax=860.8 millionths of a solar hemisphere in 

2013, smaller than either SC12 (SSAmax=1,148.9) or SC14 (SSAmax=1,195.9). Perhaps, this might be 

an indication that SSAmax (and possibly Gmax) has not yet occurred for SC24 (i.e., it might occur in 

2014 or later).
4

SSA (in millionths of a solar hemisphere)

Total Corrected Sunspot Area (1875–2013)

500

0

1,000

1,500

2,000

2,500

3,000

3,500

5

10

15

200

150

100

50

0

0

mean = 838.8

sd = 745.0

1920 1940 1960 1980 2000 2020

Year

1880 1900

(c)

(b)

(a)

G

Sunspot Number (1875–2013)

Number of Groups (1875–2013)

11 12 13 14 15 16 17 18 19 20 21 22 23 24

mean = 4.74

sd = 3.69

SSN

mean = 54.8

sd = 45.0

Figure 1. Annual variation of (a) SSN, (b) G, and (c) SSA for the interval 1875–2013.
5

Table 1. Minimum and maximum parametric values and epochs of occurrence, SC12–SC24.

Cycle SSNmin (yr) SSNmax (yr) ASC PER Gmin (yr) Gmax (yr) SSAmin (yr) SSAmax (yr) NSDmax (yr)

12 3.4 (1878) 63.7 (1883) 5 11 0.22 (1878) 5.57 (1884) 22.2 (1878) 1,148.9 (1883) 280 (1878)

13 6.3 (1889) 85.1 (1893) 4 12 0.52 (1889) 8.42 (1893) 76.7 (1889) 1,460.6 (1893) 212 (1889)

14 2.7 (1901) 63.5 (1905) 4 12 0.22 (1901) 5.35 (1907) 27.9 (1901) 1,195.9 (1905) 287 (1901)

15 1.4 (1913) 103.9 (1917) 4 10 0.17 (1913) 9.62 (1917) 7.5 (1913) 1,533.9 (1917) 311 (1913)

16 5.8 (1923) 77.8 (1928) 5 10 0.67 (1923) 7.14 (1928) 54.7 (1923) 1,388.9 (1928) 200 (1923)

17 5.7 (1933) 114.4 (1937) 4 11 0.62 (1933) 10.15 (1937) 91.3 (1933) 2,072.8 (1937) 240 (1933)

18 9.6 (1944) 151.6 (1947) 3 10 1.01 (1944) 11.78 (1947) 124.7 (1944) 2,634.1 (1947) 159 (1944)

19 4.4 (1954) 190.2 (1957) 3 10 0.45 (1954) 13.74 (1957) 34.6 (1954) 3,048.5 (1957) 241 1954)

20 10.2 (1964) 105. 9 (1968) 4 12 1.07 (1964) 9.29 (1967) 53.9 (1964) 1,601.3 (1970) 112 (1964)

21 12.6 (1976) 155.4 (1979) 3 10 1.16 (1976) 14.90 (1979) 166.4 (1975) 2,283.3 (1982) 105 (1976)

22 13.4 (1986) 157.6 (1989) 3 10 1.10 (1986) 13.06 (1990) 124.7 (1986) 2,579.2 (1989) 129 (1986)

23 8.6 (1996) 119.6 (2000) 4 12 0.83 (1996) 9.81 (2000) 81.9 (1996) 1,828.7 (2002) 165 (1996)

24 2.9 (2008) 64.9 (2013)? 5? – 0.33 (2008) 5.80 (2013)? 22.8 (2008) 860.8 (2013)? 265 (2008)

Figure 2 compares SC24 against the SC12–SC23 parametric averages of (a) <SSN>, (b) <G>, 

(c) <SSA>, and (d) <NSD> for t=–3 to 10 years relative to SSNmin occurrence. The vertical bars 

represent the mean ±1 sd interval for each t. Relative to the means, SC24 is running slightly more 

than 1 sd lower than the means of <SSN>, <G>, and <SSA>. For NSD (at t=0) SC24 is about 

62 days longer than the mean of SC12–SC23; however, as shown in table 1, NSD (at t=0) was longer 

for SC12, SC14, and SC15 than it was for SC24. From figure 2, one expects SC24 to begin decreasing in amplitudes (i.e., SSN, G, and SSA), probably beginning in 2014, with the next cycle minimum 

expected probably about 2018–2020. The first spotless day post maximum amplitude (i.e., SSNmax) 

can occur any time, especially, at t ≥ 6 years, heralding the demise of the current ongoing cycle and 

anticipating the onset of the next cycle.
6

<NSD>

50

100

150

200

250

300

(d)

–4 –3 –2 –1 0 1 2 3 4

SC24

5 6 7 8 9 10

t (Elapsed Time in Years From SSNmin)

mean SC12–SC23

mean SC12–SC23

mean SC12–SC23

mean SC12–SC23

<SSA>

500

1,000

1,500

2,000

(c)

SC24

SC24

SC24

2,405.3 2,292.2 2,353.1

12.40 12.06 11.41

150.5

150.3 139.6

<G>

5

10

(b)<SSN>

60

0

0

0

0

120

(a)

Figure 2. SC24 values of (a) SSN, (b) G, (c) SSA, and (d) NSD 

in relation to the mean of cycles 12–23 for elapsed time 

t=–3 to 10 years from SSNmin.
7

Figure 3 displays the ratios (a) SSN/G, (b) SSA/G, and (c) SSA/SSN for the interval 1875–

2013. The means of the ratios are 11.38 (±1.83), 158.6 (±40.3), and 14.01 (±3.19), respectively. For 

SSN/G, the highest ratios occur prior to 1882 (the year when the k factor was changed from 1 to 0.6), 

suggesting perhaps that the SSN values during the early interval 1875–1881 might be slightly too 

high (≤5 units of SSN). If true, then the early ratios of SSA/SSN could be slightly too low. SSA/SSN

5

0

10

15

20

mean = 14.01

sd = 3.19

1920 1940 1960 1980 2000 2020

Year

1880 1900

(c)SSA/G

100

50

150

200

250

mean = 158.6

sd = 40.3

(b)SSN/G

10

5

15

20

mean = 11.38

sd = 1.83

(a)

Figure 3. Annual variation of the ratios (a) SSN/G, (b) SSA/G, 

and (c) SSA/SSWN for the interval 1875–2013.
8

Close inspection of figure 3 suggests that the ratios might not be randomly distributed. As 

an example, for the interval 1875–2013, the median value for SSN/G is 11.27, with the number of 

values greater than or equal to the median being 71 and the number below the median being 68 and 

with 23 larger-than-median runs. Runs testing27 yields the normal deviate for the sample results as 

z=4.14, and by hypothesis testing, one surmises that there is >99.9% probability that SSN/G varies 

nonrandomly. Hence, simple relationships, such as SSN/G=11.38 (±1.83), SSA/G=158.6 (±40.3), 

or SSA/SSN=14.01 (±3.19), while approximately true, are inadequate for calculating SSN directly 

from G or SSA. (In many texts, the oft stated relation between sunspot area and sunspot number is 

A=16.7 R, where A refers to the area of sunspots and R refers to sunspot number.)

Figure 4 depicts variations of the ratios (a) <SSN/G>, (b) <SSA/G>, and (c) <SSA/SSN for 

t =–3 to 10 years relative to SSNmin occurrence (a la fig. 2). Clearly, the ratios vary over the SC with 

a minimum near cycle minimum (t=0) and a maximum near cycle maximum (t=3–6). While SC24 

has SSN/G ratio values within the mean ±1sd interval, for SSA/G and SSA/SSN SC24 ratios appear 

to now be running outside-low the mean ±1 sd intervals. Figure 4 suggests that the ratios possibly 

have all peaked at t=4 (the year 2012) and are now destined to decline with the passage of time. (In 

contrast, individual parameters SSN, G, and SSA possibly have peaked at t=5, the year 2013, as 

shown in fig. 2.)
9

<SSA/SSN>

10

5

15

20

(c)

–4 –3 –2 –1 0 1 2 3 4

SC24

5 6 7 8 9 10

t (Elapsed Time in Years From SSNmin)

<SSA/G>

100

50

150

200

(b)

SC24

<SSN/G>

10

15

(a)

SC24

210.7 221.2 213.5

mean SC12–SC23

mean SC12–SC23

mean SC12–SC23

Figure 4. SC24 values of the ratios (a) SSN/G, (b) SSA/G, 

and (c) SSA/SSN in relation to the mean of cycles 

12–23 for elapsed time t=–3 to 10 years from 

SSNmin. 
10

Figure 5 displays cyclic averages of (a) <SSN>, (b) <G>, (c) <SSA>, (d) <SSN/G>, 

(e) <SSA/G>, and (f) <SSA/SSN> for SC12–SC23. From the preceding, it is apparent that SC24 will 

have lower values than those of SC23, indicating a continuing downward progression in the parametric values with time. <SSA/SSN>

10

15

20

(f)<SSA/G>

100

150

200

(e)

12 14 16 18 20 22 24

SC

<SSN/G>

10

15

(d)<SSA>

500

1,000

1,500

(c)<G>

5

10

(b)<SSN>

50

0

0

0

100

(a)

Figure 5. SC-mean values of (a) SSN, (b) G, (c) SSA, (d) SSN/G, 

(e) SSA/G, and (f) SSA/SSN for SC12–SC23. 
11

Figure 6 compares SC24 and SC14 in terms of (a) SSN, (b) SSA, (c) G, (d) SSN/G, (e) SSA/

SSN, and (f) SSA/G. For SSN, SSA, and G the values are very close for both cycles. However, for 

the ratios, there are obvious differences. For example, while values of SSN/G appear to be trending 

downwards throughout SC14, for SC24 they appear to be more cyclic in appearance (i.e., rising and 

flattening). For the other ratios, SC14 tends to be the larger cycle and the ratios are more cyclic in 

appearance (for both SC14 and SC24).

G

5

10

(c)

–2 0 2 4 6 8 10

SC24

SC14

t

–2 0 2 4 6 8 10

t

SSA

500

0

0

1,000

1,500

(b)

SC24

SC14

SSA/SSN

10

5

15

20

(e)

SC24

SC14

SSA/G

100

50

150

200

250

(f)

SC24

SC14

SSN

20

0

40

60

80

(a)

SC24

SC14

SSN/G

10

5

15

(d)

SC14

Figure 6. SC24 values of (a) SSN, (b) G, (c) SSA, (d) SSN/G, (e) SSA/G, and (f) SSA/SSN 

in relation to SC-mean values (SC12–SC23) for elapsed time t=0–10 years. 
12

Table 2 compares the parametric means for SC14 and SC24 during the first 6 years of each 

cycle. The only parameter that has a possible significant difference, on the basis of the t statistic for 

independent samples,28 is the one for SSN/G, which has t=–2.3, inferring that the difference in the 

means for the two cycles is statistically important at the 5% level of significance.

Table 2. Results of statistical testing of the parametric means 

based on the first 6 years of SC14 and SC24 using 

the t statistic for independent samples.

Parameter SC14 SC24 t

SSN 31.9 (25.3) 33.4 (29) –0.10

SSA 480.9 (447.4) 445.2 (399) 0.15

G 2.81 (2.26) 3.00 (2.53) –0.14

SSN/G 11.87 (1.01) 10.42 (1.17) 2.30

SSA/G 159.6 (35.5) 124.1 (39.7) 1.63

SSA/SSN 13.50 (3.02) 11.67 (2.7) –1.11

Figure 7 shows the scatter plots of (a) SSN versus G and (b) SSA versus G. For SSN versus G, the inferred preferential linear regression is y=–2.019+11.993x, where y is SSN and x is G. 

The inferred regression has a coefficient of linear regression r=0.984, a coefficient of determination 

(a measure of the amount of variance explained by the regression) r 2=0.968, a standard error of 

estimate se=8.049, and a confidence level cl »99.9%. For SSA versus G, the inferred preferential 

linear regression is y=–90.988+196.099x, having r=0.971, r 2=0.943, se=178.119, and cl»99.9%. 

As an example, given G=10, one expects SSN=117.9±8 and SSA=1,870±178.1, these being the 

±1 se prediction intervals.
13

SSA

y

500

0

1,000

1,500

2,000

SSA vs. G

1875–2013

SSN vs. G

1875–2013

2,500

3,000

3,500

G

0 5 10 15

(b)

(a)

y = –90.988 + 196.099x

r = 0.971, r

2

 = 0.943

se = 178.119, cl » 99.9%

SSN

y

50

0

100

150

200

y = –2.019 + 11.993x

r = 0.984, r

2

 = 0.968

se = 8.049, cl » 99.9%

1957

1957

1978

1978

1979

1979

Figure 7. Scatter plot of (a) SSN versus G and (b) SSA versus G

for the interval 1875–2013. 
14

In figure 7, the years 1957, 1978, and 1979 are identified because of their large deviations 

from the regression line. The year 1957 represents the highest observed annual SSN to date (=190.2, 

associated with cycle maximum for SC19). Its observed value is 4.06 se higher than what one would 

have predicted for SSN given the observed G (=13.3). Similarly, the years 1978 and 1979 (associated 

with SC21) have large deviations from the regression line, but now in the opposite sense (i.e., their 

predicted SSNs are too high). For 1978, given G=11.05, one would have expected SSN=130.5; 

however, the observed SSN measured only 92.5, a difference of –4.7 se. Similarly, for 1979, given 

G=14.9, one would have expected SSN=176.7; the observed SSN instead measured only 155.4, 

a difference of –2.6 se.

Figure 8 displays the scatter plot of SSN versus SSA. The inferred preferential linear regression is y=4.959 + 0.059x, having r=0.985, r2= 0.971, se = 9.878, and cl >>99.9%. Three years are 

also identified in it as having large deviations: 1946, 1979, and 1982. For 1946 (SC18), given SSA= 

1,823.9, one would have expected SSN=112.6; instead, the observed SSN measured 92.6, a difference of –2 se. For 1979, given SSA=2,194.5, one would have expected SSN=134.4; the observed 

SSN measured 155.4, a difference of 2.1 se. For 1982, given SSA=2,283.3, one would have expected 

SSN=139.7; the observed SSN measured 115.9, a difference of –2.4 se. SSN

y

50

0

100

150

200 SSN vs. SSA

1875–2013

SSA

0 500 1,000 1,500 2,000 2,500 3,000 3,500

y = 4.959 + 0.059x

r = 0.985, r

2

 = 0.971

se = 9.878, cl » 99.9%

1982

1979

1946

Figure 8. Scatter plot of SSN versus SSA for the interval 1875–2013.
15

Figure 9 plots the observed SSN versus the bivariate fit of SSN (parameter y) against 

both G (parameter 1) and SSA (parameter 2). The bivariate equation is SSN (predicted) 

= 0.973+5.877G+0.031SSA, having Ry12=0.990 and Sy12=6.4, an improvement over using either 

G or SSA alone.29,30 SSN (Observed)

50

0

100

150

200 Observed vs. Predicted

1875–2013

1:1 Line

SSN (Predicted, G and SSA)

0 50 100 150 200

SSN (Predicted) = 0.973 + 5.877G + 0.031SSA

Ry12 = 0.990, Sy12 = 6.4

Figure 9. Scatter plot of SSN (observed) versus SSN (predicted, G and SSA) 

for the interval 1875–2013.

Figure 10 plots the difference (observed – predicted) for SSN based on using the bivariate 

equation for estimating SSN. Large discrepancies are still apparent, especially for the interval 1978–

1982 associated with SC21 (which is also during the changeover interval from Zürich SSN to SIDC 

SSN). The only years having discrepancies >±2 se (i.e., ≥±16.8) include 1957 (too high), 1978 (too low), 

1980 (too high), and 1982 (too low), with the worst difference being the one for 1980 (observed – predicted=23, a difference of 3.6 se). For SC24, all differences have measured –0.7 to 3.2 (or ≤±0.5 se).
16

Difference (Observed–Predicted)

–15

–20

–10

–5

0

5

10

15

20

25

+2 se

–2 se

+1 se

–1 se

Difference (Observed–Predicted)

1875–2013

1920 1940 1960 1980 2000 2020

Year

1880 1900

Figure 10. Annual variation of the difference (observed – predicted) 

for the interval 1875–2013.

Interestingly, it appears that the differences were trending downwards from positive values to 

negative values prior to about 1930, but were trending upwards towards positive values thereafter 

until about 1960. While there are about equal numbers of positive and negative differences, runs testing suggests that the overall behavior of the differences is nonrandom in nature. About 80% of the 

differences lie within the ±1 se band (i.e., the difference is with ±1 se, or within 6.4 units of sunspot 

number). Table 3 provides a direct accounting of the observed and predicted (based on G and SSA) 

values and the difference (observed – predicted).
17

Table 3. Comparison of observed and predicted SSN, where predicted SSN 

is the bivariate fit of SSN based on G and SSA.

Year Observed

Predicted

(G,SSA) Difference Year Observed

Predicted

(G,SSA) Difference Year Observed

Predicted

(G,SSA) Difference

1875 17.0 13.9 3.1 Cycle 15 1950 83.9 76.6 12.3

1876 11.3 8.6 2.7 1913 1.4m 2.2m –0.7 1951 69.4 66.1 3.3

1877 12.4 7.6 4.8 1914 9.6 12.7 –3.1 1952 31.5 28.9 2.6

Cycle 12 1915 47.4 52.6 –5.2 1953 13.9 12.4 1.5

1878 3.4m 3.0m 0.4 1916 57.1 60.3 –3.2 Cycle 19

1879 6.0 4.0 2.0 1917 103.9M 105.1M –1.2 1954 4.4m 4.7m –0.3

1880 32.3 27.3 5.0 1918 80.6 82.2 –1.6 1955 38.0 37.1 0.9

1881 54.3 45.5 9.8 1919 63.6 69.9 –6.3 1956 141.7 136.6 5.1

1882 59.7 57.1 2.6 1920 37.6 42.5 –4.9 1957 190.2M 173.6 16.6

1883 63.7M 65.4 –1.7 1921 26.1 29.1 –3.0 1958 184.8 175.1M 9.7

1884 63.5 65.8M –2.3 1922 14.2 17.5 –3.3 1959 159.0 162.7 –3.7

1885 52.2 50.7 1.5 Cycle 16 1960 112.6 104.2 8.4

1886 25.4 24.8 0.6 1923 5.8m 6.6m –0.8 1961 53.9 45.6 8.3

1887 13.1 8.3 4.8 1924 16.7 19.1 –2.4 1962 37.6 31.9 5.7

1888 6.8 7.9 –1.1 1925 44.3 54.2 –9.9 1963 27.9 23.1 4.8

Cycle 13 1926 63.9 75.7 –11.8 Cycle 20

1889 6.3m 6.4m –0.1 1927 69.0 72.0 –3.0 1964 10.2m 8.9m 1.3

1890 7.1 8.2 –1.1 1928 77.8M 86.0M –8.2 1965 15.1 11.7 3.4

1891 35.6 38.7 –3.1 1929 64.9 75.3 –8.4 1966 47.0 43.4 3.6

1892 73.0 75.8 –2.8 1930 35.7 39.4 –3.7 1967 93.8 102.7 –8.9

1893 85.1M 95.7M –10.6 1931 21.2 22.2 –1.0 1968 105.9M 97.2 8.7

1894 78.0 84.8 –6.8 1932 11.1 13.6 –2.5 1969 105.5 93.1 12.4

1895 64.0 65.5 –1.5 Cycle 17 1970 104.5 105.0M –0.5

1896 41.8 36.9 4.9 1933 5.7m 7.4m – 1.7 1971 66.6 67.6 –1.1

1897 26.2 31.4 –5.2 1934 8.7 10.0 –1.3 1972 68.9 65.7 3.2

1898 26.7 25.1 1.6 1935 36.1 41.0 –4.9 1973 38.0 34.6 3.4

1899 12.1 10.6 1.5 1936 79.7 79.8 –0.1 1974 34.5 31.0 3.5

1900 9.5 7.7 1.8 1937 114.4M 124.9M –10.5 1975 15.5 13.8 1.7

Cycle 14 1938 109.6 118.2 –8.6 Cycle 21

1901 2.7m 3.1m –0.4 1939 88.8 96.4 –7.8 1976 12.6m 13.1m –0.5

1902 5.0 5.0 – 1940 67.8 67.2 0.6 1977 27.5 26.7 0.8

1903 24.4 23.7 0.7 1941 47.5 45.1 2.4 1978 92.5 108.3 –15.8

1904 42.0 38.1 3.9 1942 30.6 30.6 – 1979 155.4M 151.3M 4.1

1905 63.5M 69.3M –5.8 1943 16.3 18.2 –1.9 1980 154.6 131.6 23.0

1906 53.8 55.0 –1.2 Cycle 18 1981 140.4 134.5 5.9

1907 62.0 66.3 –4.3 1944 9.6m 10.8m –1.2 1982 115.9 132.2 –16.3

1908 48.5 49.7 –1.2 1945 33.2 31.9 1.3 1983 66.6 69.7 –3.1

1909 43.9 45.6 –1.7 1946 92.6 103.2 –10.6 1984 45.9 49.8 –3.9

1910 18.6 21.6 –3.0 1947 151.6M 151.9M –0.3 1985 17.9 16.3 1.6

1911 5.7 7.4 –1.7 1948 136.3 124.7 11.6

1912 3.6 4.6 –1.0 1949 134.7 130.1 4.6
18

Table 3. Comparison of observed and predicted SSN, where predicted SSN 

is the bivariate fit of SSN based on G and SSA (Continued).

Year Observed

Predicted

(G,SSA) Difference Year Observed

Predicted

(G,SSA) Difference Year Observed

Predicted

(G,SSA) Difference

Cycle 22 Cycle 23 Cycle 24

1986 13.4m 11.3m 2.1 1996 8.6m 8.4m 0.2 2008 2.9m 3.6m –0.7

1987 29.4 25.3 4.1 1997 21.5 18.6 2.9 2009 3.1 3.8 –0.7

1988 100.2 89.6 10.6 1998 64.3 55.8 8.5 2010 16.5 16.7 –0.2

1989 157.6M 157.2M 0.4 1999 93.3 81.0 12.3 2011 55.6 53.1 2.5

1990 142.6 141.2 1.4 2000 119.6M 108.7 10.9 2012 57.5 55.4 2.1

1991 145.7 153.4 –7.7 2001 111.0 109.7 1.3 2013 64.9M 61.7M 3.2

1992 94.3 91.9 2.4 2002 104.0 114.3M –10.4

1993 54.6 48.9 5.7 2003 63.7 69.5 –5.8

1994 29.9 29.1 0.8 2004 40.4 43.2 –2.8

1995 17.5 16.3 1.2 2005 29.8 33.5 –3.7

2006 15.2 10.6 4.6

2007 7.5 10.0 –2.5
19

3. DISCUSSION AND SUMMARY

The oldest routinely observed solar index (1826–1868) for measuring activity on the Sun is 

Schwabe’s ‘clusters of spots’ (essentially, G) and the ‘number of days when no spots were observed’ 

(or NSD).1 Schwabe’s simple method for monitoring the spottiness on the Sun was later superseded 

by the use of Wolf’s relative sunspot number,6 an index that he introduced in 1848 (based on G 

and N) and used to reconstruct the sunspot record back to 1610, when Galileo first reported seeing spots on the Sun. About 1882, however, a change7 was made in the methodology for counting 

sunspots (i.e., the inclusion of smaller spots and the use of a weighting scheme) at the Swiss Federal 

Observatory such that a factor k=0.6 was introduced to bring measurements of sunspot number 

into agreement with Wolf’s earlier values (which neglected smaller spots). Then, in 1981, the Royal 

Observatory of Belgium9 (i.e., the SIDC) took over the task of determining sunspot number values 

from the Swiss Federal Observatory and continues to do so today, employing a variable k (equal to 

about 0.9, on average, being dependent upon the actual number of observers10).

Examined in this study has been the annual variations and ratios of SSN, G, and SSA for 

the overall interval 1875–2013, correlations of SSN against G and SSA, and the correlation of SSN 

(observed) against the bivariate fit based on the combined variations of G and SSA. Also, parametric 

variations during SC24 have been contrasted with the mean values based on SC12–SC23 and SC14, 

the smallest sunspot cycle of the modern era.

While SSN is found to independently correlate strongly (r = 0.98) against G (y=–2+11.99x) 

and SSA (y=5+0.059x), a stronger correlation (Ry12=0.99) is the one based on the bivariate fit of 

SSN against the combined variations of G and SSA (y=1+5.88x1+0.031x2). Also, based on the 

bivariate fit, while all cycle minima are found to occur concurrently with SSNmin, cycle maxima 

sometimes are found to occur at different times. For example, based on the inferred bivariate fit, the 

predicted SSNmax is inferred to have occurred in 1884, 1958, 1970, and 2002, respectively, for SC12, 

SC19, SC20, and SC23 rather than in 1883, 1957, 1968, and 2000, based on the observed SSNmax. 

Also, based on the bivariate fit, the observed SSNmax for SC19 seems too high. During the 139-year 

interval 1875–2013, the difference between the observed and predicted SSN based on the bivariate fit is <1 Sy12 (<6.4) for 111 years, between 1 and <2 Sy12 (6.4 –<12.8) for 28 years and ≥2 Sy12

(≥12.8) for only 4 years, these particular years being 1957 (16.6), 1978 (–15.8), 1980 (23), and 1982 

(–16.3). For sunspot cycle 24 the difference between observed and predicted SSN has been –0.7 to 3.2 

(≤0.5 Sy12).

In conclusion, had Wolf used SSA or G and SSA rather than G and N for describing solar 

activity, the record would look slightly different, especially as related to the amplitudes and occurrences of SSNmax for the years 1957, 1978 1980, and 1982.
20
21

REFERENCES

 1. Schwabe, H.: “Sonnen-Beobachtungen im Jahre 1843,” Astron. Nach., Vol. 21, No. 495, pp. 233–

236, 1844.

 2. Wilson, R.M.: “A Comparison of Wolf’s Reconstructed Record of Annual Sunspot Number 

with Schwabe’s Observed Record of ‘Clusters of Spots’ for the Interval of 1826-1868,” Solar 

Phys., Vol. 182, pp. 217–230, 1998.

3.	Wilson, R.M.: Volcanism, Cold Temperature, and Paucity of Sunspot Observing Days (1818-1858):

A Connection? NASA/TP—1998–208592, Marshall Space Flight Center, AL, 36 pp., August 1998.

 4. Complete Dictionary of Scientific Biography, <http://www.encyclopedia.com/topic/Samuel_Heinrich_Schwabe.aspx>, 2008.

5. Kosmos. Entwurf einer physischen Weltbeschreibung von Alexander von Humboldt, <http://www.biodiversitylibrary.org/bibliography/4717>, five volumes, 2008.

6.	Kiepenheuer, K.O.: “Ch. 6. Solar Activity,” in The Sun, G.P. Kuiper (ed.), The University of Chicago

Press, Chicago, pp. 322–465, 1953.

7.	Waldmeier, M.: The Sunspot-Activity in the Years 1610-1960, Schulthess & Co., Zürich, Switzerland,

171 pp., 1961.

8.	 Izenman, A.J.: “J. R. Wolf and the Zürich Sunspot Relative Numbers,” The Math. Intelligencer,

Vol. 7, No. 1, pp. 27–33, 1985.

9.	Clette, F.; Berghmans, D.; Vanlommel, P.; et al.: “From the Wolf Number to the International Sunspot

Number Index: 25 Years of SIDC,” Adv. Space Res., Vol. 40, pp. 919–928, 2007.

10.	Feehrer, C.E.: “Dances with Wolf’s: A Short History of Sunspot Indices,” <http://www.aavso.org/

dances-wolfs-short-history-sunspot-indices>, August 2000.

11.	 Hoyt, D.V.; and Schatten, K.H.: The Role of the Sun in Climate Change, Oxford University Press, New

York, 279 pp., 1997.

12.	Hoyt, D.V.; and Schatten, K.H.: “Group Sunspot Numbers: A New Solar Activity Reconstruction,”

Solar Phys., Vol. 179, pp. 189–219, 1998.

13.	Hathaway, D.H.; Wilson, R.M.; and Reichmann, E.J.: “Group Sunspot Numbers: Sunspot Cycle

Characteristics,” Solar Phys., Vol. 211, pp. 357–370, 2002.
22

14.	Hathaway, D.H.: “The Solar Cycle,” Living Rev. Solar Phys., Vol. 7, 65 pp., <http://www.livingreviews.org/lrsp-2010-1>, 2010.

15.	Leussu, R.; Usoskin, I.G.; Arlt, R.; and Mursula, K.: “Inconsistency of the Wolf Sunspot Number

Series Around 1848,” Astron. & Astrophys., Vol. 559, p. A28, 4 pp., November 2013.

16.	Wilson, R.M.; and Hathaway, D.H.: A Comparison of Rome Observatory Sunspot Area and Sunspot 

Number Determinations with International Measures, 1958-1998, NASA/TP—2005–214191, Marshall Space Flight Center, AL, 20 pp., November 2005.

17.	Wilson, R.M.; and Hathaway, D.H.: On the Relation between Sunspot Area and Sunspot Number,

NASA/TP—2006–214324, Marshall Space Flight Center, AL, 20 pp. February 2006.

18. “Royal Observatory, Greenwich—USAF/NOAA Sunspot Data,” Solar Physics, NASA Marshall

Space Flight Center, Huntsville, AL, <http://solarscience.msfc.nasa.gov/greenwch.shtml>, March 3,

2014.

19. Wilson, R.M.: “On the Use of ‘First Spotless Day’ as a Predictor for Sunspot Minimum,” Solar 

Phys., Vol. 158, pp. 197–204, 1995.

20. Wilson, R.M.; Hathaway, D.H.; and Reichmann, E.J.: “On the Behavior of the Sunspot Cycle 

near Minimum,” J. Geophys. Res., Vol. 101, No. A9, pp. 19,967–19,972, September 1996.

21. Wilson, R.M.; Hathaway, D.H.; and Reichmann, E.J.: Gauging the Nearness and Size of Cycle 

Minimum NASA Technical Paper 3674, Marshall Space Flight Center, AL, 20 pp., February 

1997. 

22. Harvey, K.L.; and White, O.R.: “What is Solar Cycle Minimum?” J. Geophys. Res., Vol. 104, No. 

A9, pp. 19,759–19,764, September 1999.

23. Wilson, R.M.; and Hathaway, D.H.: On the Relation Between Spotless Days and the Sunspot 

Cycle, NASA/TP—2005–213608, Marshall Space Flight Center, AL, 32 pp., January 2005.

24. Wilson, R.M.; and Hathaway, D.H.: On the Relationship between Spotless Days and the Sunspot Cycle: A Supplement, NASA/TP—2006–214601, Marshall Space Flight Center, AL, 64 pp., 

August 2006.

25. Russell, C.T.; Luhmann, J.G.; and Jian, L.K.: “How Unprecedented a Solar Minimum?” Rev. 

Geophys., Vol. 48, p. RG2004, 16 pp., 2010.

26. Wilson, R.M.; Hathaway, D.H.; and Reichmann, E.J.: On Determining the Rise, Size, and Duration Classes of a Sunspot Cycle, NASA Technical Paper 3652, Marshall Space Flight Center, AL, 

14 pp., September 1996.
23

27. Lapin, L.L.: Statistics for Modern Business Decisions, 2nd ed., Harcourt Brace Jovanovich, Inc., 

New York, p. 626, 1978.

28. Lapin, L.L.: Statistics for Modern Business Decisions, 2nd ed., Harcourt Brace Jovanovich, Inc., 

New York, p. 486, 1978.

29. Ehrenberg, A.S.C.: A Primer in Data Reduction, John Wiley & Sons, New York, p. 200, 1982.

30. Lapin, L.L.: Statistics for Modern Business Decisions, 2nd ed., Harcourt Brace Jovanovich, Inc., 

New York, p. 377, 394, 1978.
24

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Comparison of the Variations of Sunspot Number, Number

of Sunspot Groups, and Sunspot Area, 1875–2013

Robert M. Wilson

George C. Marshall Space Flight Center

Huntsville, AL 35812

National Aeronautics and Space Administration

Washington, DC 20546–0001

Unclassified-Unlimited

Subject Category 92

Availability: NASA STI Information Desk (757–864–9658)

Prepared by the Science and Research Office, Science and Technology Office

M–1381

Technical Publication

NASA/TP—2014–218192

sunspot cycle, sunspot number, sunspot area, number of sunspot groups

01–04–2014

UU 36

NASA

U U U

Examined are the yearly variations and ratios of sunspot number, the number of sunspot groups, and the total corrected sunspot area for the interval 1875–2013. 

While yearly sunspot number independently correlates strongly (r=0.98) with the yearly number of sunspot groups (y=–2+11.99x) and the total corrected sunspot 

area (y= 5+0.059x), the strongest correlation (Ry12=0.99) is the one based on the bivariate fit of sunspot number against the combined variations of the number of 

sunspot groups and sunspot area (y=1+5.88x1+0.031x2, where y refers to sunspot number, x1 refers to the number of sunspot groups, and x2 refers to the sunspot 

area). While all cycle minima based on the bivariate fit are concurrent with the observed minimum in sunspot number, cycle maxima are sometimes found to differ. 

For sunspot cycles 12, 19, 20, and 23, cycle maximum is inferred to have occurred in 1884, 1958, 1970, and 2002, respectively, rather than in 1883, 

1957, 1968, and 2000, based on the observed sunspot number. Also, cycle 19’s maximum amplitude based on observed sunspot number seems 

too high in comparison to that found using the bivariate fit. During the 139-year interval 1875–2013, the difference between the observed and 

predicted sunspot number based on the bivariate fit is <1 standard error of estimate (se) (<6.4) for 111 years, between 1 and <2 se (6.4 to <12.8) 

for 28 years, and ≥2 se (≥12.8) for only 4 years, these years being 1957 (16.6), 1978 (–15.8), 1980 (23), and 1982 (–16.3). For sunspot cycle 24, the 

difference between observed and predicted values has been only –0.7 and 3.2 (≤0.5 se).

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