How to use the VAR.S function

The VAR.S function tries to estimate the variance based on a sample of the population. The function ignores logical and text values.

The VAR.P function also calculates the variance, however, it assumes the data set is the entire population and not a sample. This makes the output differ slightly from these two functions.

1. Introduction

What is the variance measure in statistics?

The variance shows how much a set of numbers are spread out from their average value.

Variance tells you how far from the average values are spread out. Both charts above have numbers and an average plotted, they share the same average however, the numbers are not the same.

Chart A above shows that the values are more spread out than the values in chart B. Chart A has a variance of approx 550.02, the variance for chart B is approx 27.11. Variance is often used in statistics.

One limitation of the variance is that its units are different from the units of the original random variable. The standard deviation retains the same units as the random variable making it a more useful measure of spread or dispersion.

A normal distribution is a symmetric, bell-shaped probability distribution that is commonly used in statistics and probability theory. The shape of the normal distribution is determined by its mean (μ) and standard deviation (σ). Standard deviation is the square root of the variance. The mean represents the center of the distribution, while the standard deviation represents the spread or dispersion of the data around the mean.

• A normal distribution with a standard deviation of 0.5 is relatively narrow and tightly clustered around the mean.
  The values in the distribution are concentrated within a smaller range, with most values falling closer to the mean. The curve appears tall and steep, indicating a higher concentration of data points near the mean.
• A normal distribution with a standard deviation of 1 is the most commonly used normal distribution, often referred to as the standard normal distribution.
  
  The standard deviation of 1 represents a moderate spread of the data around the mean. About 68% of the data falls within one standard deviation (±1σ) of the mean, and approximately 95% of the data falls within two standard deviations (±2σ) of the mean. The curve has a characteristic bell shape, with a smooth and gradual taper towards the tails.
• A normal distribution with a standard deviation of 2 is relatively wide and spread out compared to the standard normal distribution. The data is dispersed over a larger range, with values more spread out from the mean. The curve appears shorter and flatter, indicating a lower concentration of data points near the mean.

How is the variance calculated?

It depends on which function you use, the VAR.P function or the VAR.S function. The VAR.S function calculates the variance based on a sample of the population.

VAR.S function = Σ(x - x̄)²/(n-1)

The VAR.P function calculates the variance based on the population.

VAR.P function = Σ(x - x̄)²/n

x is each value
x̄ is the mean of all values
n is the total number of observations

How is the variance and standard deviation related?

The standard deviation is the square root of the variance. The following formula shows how the standard deviation is calculated.

STDEV.S function = √(Σ(x - x̄)²/(n-1))

VAR.S function = Σ(x - x̄)²/(n-1)

2. VAR.S Function Syntax

VAR.S(number1,[number2],...)

3. VAR.S Function Arguments

number1	Required.  A cell reference to the sample of the population.
number2	Optional. Up to 254 additional arguments.

4. VAR.S Function Example

Your organization has collected data on the commute duration for its employees. To gain insight into the variability of commute times among the workforce, calculate the sample variance of the commute time values? This statistical measure will provide an understanding of how dispersed or concentrated the commute durations are within the sample of employees surveyed.

The data points are in cell range B16:B25, here they are: 49, 19, 33, 15, 36, 38, 17, 23, 43, and 18

The argument in this example is:

• number1 = B16:B25

There are 10 data points in this example.

Formula in cell E15:

Cell E15 returns 148.77 which represents the variance. The standard deviation is the square root of the variance. √148.77 equals 12.20 which is the same value that the STDEV.S function returns.

The image above shows a chart containing a blue line that represents the normal distribution based on a mean of 29 and a standard deviation of 12.20. The chart also shows the different standard deviations 1σ, 2σ, 3σ, -1a, -2σ, and -3σ which represents:

• 68% of the data falls between μ ± 1σ
• 95% of the data falls between μ ± 2σ
• 99.7% of the data falls between μ ± 3σ

5. How is the VAR.S function calculated?

The equation for VAR.S is:

x ̅  is the sample mean AVERAGE(number1,number2,…)

n is the sample size.

Using the example above (Set1), the average of 10, 30, 25, 50 and 35 is 30.

(10-30)^2+(30-30)^2+(25-30)^2+(50-30)^2+(35-30)^2 = 850

850 / 4 = 212.5.

6. What is the difference between the standard deviation and variance?

Standard deviation is the square root of the variance. Both standard deviation and variance describes how spread out data is, but standard deviation is more commonly used because it is easier to relate to the data points.

For example, if the average weight of a group of people is 140 pounds and the standard deviation is 40 pounds, you can conclude that most people are within 40 pounds of the average weight. Variance, on the other hand, would be 1600, which is harder to interpret. However, variance is easier to calculate.

7. Sort rows by variance based on a sample of a population

This example demonstrates a formula in cell B8 that calculates the variance of a sample per row. It then sorts the rows from cell range B3:N6 by the variance of a sample from large to small. Cell ranges P3:P6 and P7:P11 contains the variances.

Excel 365 formula in cell B8:

The image above shows the source data in cell range B3:N6, the variances are in column P, however, they are only there to show that the calculations are correct.

The destination cell is in B8 and the formula returns an array of values to cells below and to the right as far as needed. These values are sorted based on their variances per row, these values are displayed in cell P8:P11. Note, these values are not required for the formula to work properly. The formula needs only the source data to automatically calculate the variances.

Explaining formula

The Evaluate Formula tool is located on the Formulas tab in the Ribbon. It is a useful feature that allows you to step through and evaluate complex formulas to understand how the calculation is being performed and identify any errors or issues. The following steps shows these detailed evaluations for the formula above.

Step 1 - Calculate the variance of a sample

VAR.S(a)

Step 2 - Build the LAMBDA function

The LAMBDA function build custom functions without VBA, macros or javascript.

Function syntax: LAMBDA([parameter1, parameter2, …,] calculation)

LAMBDA(a,VAR.S(a))

Step 3 - Calculate the variance of a sample by row

The BYROW function puts values from an array into a LAMBDA function row-wise.

Function syntax: BYROW(array, lambda(array, calculation))

BYROW(C3:N6,LAMBDA(a,VAR.S(a)))

returns

{51514.0606060606; 88506.5151515151; 95454.2424242424; 45039.4242424243}

Step 4 - Sort rows based on the variance of a sample

The SORTBY function sorts a cell range or array based on values in a corresponding range or array.

Function syntax: SORTBY(array, by_array1, [sort_order1], [by_array2, sort_order2],…)

SORTBY(B3:N6,BYROW(C3:N6,LAMBDA(a,VAR.S(a))),-1)

becomes

SORTBY(B3:N6,{51514.0606060606;88506.5151515151;95454.2424242424;45039.4242424243},-1)

and returns

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Functions in 'Statistical' category

The VAR.S function function is one of 73 functions in the 'Statistical' category.

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