Various equations to calculate variance and standard deviation
Chapter 3 Descriptive Statistics
Variance and Standard Deviation Calculations
Didactic and Calculating Equations In the textbook we present various equations for calculating the two most common statistics to express the amount of dispersion in a set of scores (variance and standard deviation). The definitional equations are presented to illustrate exactly what values are involved in determining these two statistics. Unfortunately these didactic equations are not very user friendly.
Population vs. Sample Equations We next point out that there are different equations for these two statistics depending on if the data are considered to be a population or a sample from a population. As explained in the textbook the difference is that for the variance of a population the denominator is N but for the sample variance the denominator is N-1. The equation for the standard deviation is the square root of the variance for both populations and for samples.
Didactic Equations Variance
Population
Sample
2
𝜎 =
2
𝑠 =
𝑋𝑖 −𝜇 2 𝑁
𝑋𝑖 −𝑋 2 𝑁−1
Standard deviation
𝜎=
𝑠=
𝑋𝑖 − 𝜇
2
𝑁
𝑋𝑖 − 𝑋 𝑁−1
Notice the equations on the right are the square root of the one on the left because the standard deviation is the square root of the variance.
2
Calculating Equations Variance
Population
Sample
Standard deviation
𝑋 𝑁
2
𝜎2 =
𝑋 −
𝑁
2
𝑠2 =
𝑋 − 𝑁−1
𝑋 𝑁
2
2
𝜎=
𝑋 −
2
𝑋 𝑁
2
𝑁
2 2
𝑠=
𝑋 𝑁
𝑋 − 𝑁−1
Notice the equations on the right are the square root of the one on the left because the standard deviation is the square root of the variance.
What does SPSS assume the data to be? Let’s use the same small data set we used in the text to illustrate the calculations of the variance and standard deviations to determine this. Recall our distribution was: 1 2 3 4 5
SPSS data Screen
SPSS Calculation of Variance and Standard Deviation
SPSS Assumes the Data represent a Sample SPSS is using equation 3.10 to calculate the variance of the data set to be 2.5 and the standard deviation to be 1.58. SPSS ASSUMES that your data consists of a SAMPLE, thus N-1 is used in the denominator to calculate Variance and Standard Deviation
Variance and Standard Deviation Calculations
Didactic and Calculating Equations In the textbook we present various equations for calculating the two most common statistics to express the amount of dispersion in a set of scores (variance and standard deviation). The definitional equations are presented to illustrate exactly what values are involved in determining these two statistics. Unfortunately these didactic equations are not very user friendly.
Population vs. Sample Equations We next point out that there are different equations for these two statistics depending on if the data are considered to be a population or a sample from a population. As explained in the textbook the difference is that for the variance of a population the denominator is N but for the sample variance the denominator is N-1. The equation for the standard deviation is the square root of the variance for both populations and for samples.
Didactic Equations Variance
Population
Sample
2
𝜎 =
2
𝑠 =
𝑋𝑖 −𝜇 2 𝑁
𝑋𝑖 −𝑋 2 𝑁−1
Standard deviation
𝜎=
𝑠=
𝑋𝑖 − 𝜇
2
𝑁
𝑋𝑖 − 𝑋 𝑁−1
Notice the equations on the right are the square root of the one on the left because the standard deviation is the square root of the variance.
2
Calculating Equations Variance
Population
Sample
Standard deviation
𝑋 𝑁
2
𝜎2 =
𝑋 −
𝑁
2
𝑠2 =
𝑋 − 𝑁−1
𝑋 𝑁
2
2
𝜎=
𝑋 −
2
𝑋 𝑁
2
𝑁
2 2
𝑠=
𝑋 𝑁
𝑋 − 𝑁−1
Notice the equations on the right are the square root of the one on the left because the standard deviation is the square root of the variance.
What does SPSS assume the data to be? Let’s use the same small data set we used in the text to illustrate the calculations of the variance and standard deviations to determine this. Recall our distribution was: 1 2 3 4 5
SPSS data Screen
SPSS Calculation of Variance and Standard Deviation
SPSS Assumes the Data represent a Sample SPSS is using equation 3.10 to calculate the variance of the data set to be 2.5 and the standard deviation to be 1.58. SPSS ASSUMES that your data consists of a SAMPLE, thus N-1 is used in the denominator to calculate Variance and Standard Deviation
