Summarize:

The term "population" refers to the total set of relevant observations. For example, if you're studying the age of Texas residents, your population would include the age of every single Texas resident. You would normally create a spreadsheet for a large data set like that, but here's a smaller example data set: Example: There are exactly six fish tanks in a room of the aquarium. The six tanks contain the following numbers of fish:x1=5{\displaystyle x_{1}=5}x2=5{\displaystyle x_{2}=5}x3=8{\displaystyle x_{3}=8}x4=12{\displaystyle x_{4}=12}x5=15{\displaystyle x_{5}=15}x6=18{\displaystyle x_{6}=18} Since a population contains all the data you need, this formula gives you the exact variance of the population. In order to distinguish it from sample variance (which is only an estimate), statisticians use different variables: 2{\displaystyle ^{2}} = ((xi{\displaystyle x_{i}} - )2{\displaystyle ^{2}})/n 2{\displaystyle ^{2}} = population variance. This is a lower-case sigma, squared. Variance is measured in squared units. xi{\displaystyle x_{i}} represents a term in your data set. The terms inside will be calculated for each value of xi{\displaystyle x_{i}}, then summed. is the population mean n is the number of data points in the population When analyzing a population, the symbol ("mu") represents the arithmetic mean. To find the mean, add all the data points together, then divide by the number of data points. You can think of the mean as the "average," but be careful, as that word has multiple definitions in mathematics. Example: mean = = 5+5+8+12+15+186{\displaystyle {
rac {5+5+8+12+15+18}{6}}} = 10.5 Data points close to the mean will result in a difference closer to zero. Repeat the subtraction problem for each data point, and you might start to get a sense of how spread out the data is. Example:x1{\displaystyle x_{1}} - = 5 - 10.5 = -5.5x2{\displaystyle x_{2}} - = 5 - 10.5 = -5.5x3{\displaystyle x_{3}} - = 8 - 10.5 = -2.5x4{\displaystyle x_{4}} - = 12 - 10.5 = 1.5x5{\displaystyle x_{5}} - = 15 - 10.5 = 4.5x6{\displaystyle x_{6}} - = 18 - 10.5 = 7.5 Right now, some of your numbers from the last step will be negative, and some will be positive. If you picture your data on a number line, these two categories represent numbers to the left of the mean, and numbers to the right of the mean. This is no good for calculating variance, since these two groups will cancel each other out. Square each number so they are all positive instead. Example:(xi{\displaystyle x_{i}} - )2{\displaystyle ^{2}} for each value of i from 1 to 6:(-5.5)2{\displaystyle ^{2}} = 30.25(-5.5)2{\displaystyle ^{2}} = 30.25(-2.5)2{\displaystyle ^{2}} = 6.25(1.5)2{\displaystyle ^{2}} = 2.25(4.5)2{\displaystyle ^{2}} = 20.25(7.5)2{\displaystyle ^{2}} = 56.25 Now you have a value for each data point, related (indirectly) to how far that data point is from the mean. Take the mean of these values by adding them all together, then dividing by the number of values. Example:Variance of the population = 30.25+30.25+6.25+2.25+20.25+56.256=145.56={\displaystyle {
rac {30.25+30.25+6.25+2.25+20.25+56.25}{6}}={
rac {145.5}{6}}=} 24.25 If you're not sure how this matches the formula at the beginning of this method, try writing out the whole problem in longhand: After finding the difference from the mean and squaring, you have the value (x1{\displaystyle x_{1}} - )2{\displaystyle ^{2}}, (x2{\displaystyle x_{2}} - )2{\displaystyle ^{2}}, and so on up to (xn{\displaystyle x_{n}} - )2{\displaystyle ^{2}}, where xn{\displaystyle x_{n}} is the last data point in the set. To find the mean of these values, you sum them up and divide by n: ( (x1{\displaystyle x_{1}} - )2{\displaystyle ^{2}} + (x2{\displaystyle x_{2}} - )2{\displaystyle ^{2}} + ... + (xn{\displaystyle x_{n}} - )2{\displaystyle ^{2}} ) / n After rewriting the numerator in sigma notation, you have ((xi{\displaystyle x_{i}} - )2{\displaystyle ^{2}})/n, the formula for variance.
Start with a population data set. Write down the population variance formula. Find the mean of the population. Subtract the mean from each data point. Square each answer. Find the mean of your results. Relate this back to the formula.