How to use the STDEV.S function

What is the STDEV.S function?

The STDEV.S function returns standard deviation based on a sample of a population.

1. Introduction

What is Standard Deviation?

Standard deviation tells you how far from the average values are spread out. Both charts above have numbers and an average plotted, they share the exact 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 standard deviation of 23.45256334, standard deviation for chart B is 5.207075606. Standard deviation is fundamental in statistics.

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). 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.

What is the difference between the STDEV.P function and the STDEV.S function?

STDEV.P function calculates the standard deviation for a population and the STDEV.S function calculates the standard deviation for a sample. STDEV.P uses the count of all values (n) in the denominator.
STDEV.S uses (n-1) in the denominator (Bessel's correction). This accounts for the difference between sample variance and population variance in statistics. STDEV.S is better for sample inferential statistics.

STDEV.P math formula:

STDEV.S math formula:

When to use the STDEV.P function and the STDEV.S function?

Use STDEV.P if you have the full population data. Use STDEV.S if you have a sample of limited data from a larger population. STDEV.P will result in a lower standard deviation compared to STDEV.S on the same data.

Sample standard deviation is considered a better estimate for inferring population parameters.

What is inferring population parameters?

Population parameters refer to the actual values of statistics that describe an entire population, such as the population mean or standard deviation. However, the true population parameter values are often not known.

What is sample inferential statistics?

Sample inferential statistics are methods that allow using statistics calculated on a sample of data to infer the unknown population parameters.

For example:

• The sample mean can be used to estimate the population mean.
• The sample standard deviation can estimate the population standard deviation.

How can standard deviation be used to find outliers?

Standard deviation can be used to identify potential outliers in a dataset by defining a range based on the mean and standard deviation values. Observations that fall outside this range are considered outliers.

A common approach is to use the range μ ± 3σ, which covers approximately 99.7% of the data points if the distribution is normal.

Values below (μ - 3σ) are considered potential lower outliers.
Values above (μ + 3σ) are considered potential upper outliers.

μ - the mean
σ - standard deviation

What is the 68–95–99.7 rule?

The 68–95–99.7 rule, also known as the Empirical Rule, is a useful statistical principle that describes the percentage of data values that fall within certain standard deviations from the mean in a normal distribution.

• 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

It's important to note that the 68–95–99.7 rule is an approximation and assumes that the data follows a normal distribution. In cases of skewed or heavy-tailed distributions the percentages may not hold true.  Alternative methods may be needed to describe the data spread accurately.

2. STDEV.S Function Syntax

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

3. STDEV.S Function Arguments

number1	Required. The first number argument that represents a sample of the population.
[number2]	Optional. Up to 253 additional number arguments.

The STDEV.S function ignores logical values and text values.

For large sample sizes, STDEV.S and STDEV.P return approximately equal outputs.

4. STDEV.S Function Example

A software engineer has measured app usage based on metrics from a group of users. Assume the data follows a normal distribution. Find the outliers outside the 2 standard deviations (±2σ) range?

The data is:

App usage
66
88
102
109
114
109
103
106
89
100

The argument is:

number1 = B16:B25

The data points are specified in cells B16:B25 in the image above.

The image above shows a chart containing a blue curve representing the probability mass function of a normal distribution where the mean is 99 and the standard deviation is 14.18, calculated below. The chart shows the standard deviations denoted ±1σ, ±2σ, and ±3σ, cells E19:E20 display ±2σ standard deviation. 2σ = 70.25 and -2σ = 126.95

Formula in cell E15:

Cell E15 displays 14.18 which represents the standard deviation (σ) based on the values in B15:B25. We can use this value and the arithmetic mean to calculate the ±2σ standard deviations.

Formula in cell E19:

The formula in cell E19 first calculates the arithmetic mean using the AVERAGE function. It then takes this mean value and subtracts twice the standard deviation from it. Specifically, it subtracts the product of 14.18 and 2 from 99, resulting in a value of 70.25.

99 - (14.175 × 2) = 98.6 - 28.35 = 70.25

Formula in cell E20:

The formula in cell E20 uses the AVERAGE function to calculate the arithmetic mean then calculates the total with the standard deviation multiplied by 2.

99 + (14.18 × 2) = 98.6 + 28.35 = 126.95

The chart in the image above shows a normal distribution with the standard deviations denoted ±1σ, ±2σ, and ±3σ. One data point is outside the ±2σ standard deviations which is 66.

5. How is the STDEV.S Function calculated?

This example demonstrates how to manually calculate the standard deviation based on a sample. The image above shows data points in cell range B3:B12, the calculation is in cell D3. Here is the data:

Number
137
139
141
105
139
124
126
146
105
101

The math formula behind the calculation in cell D3 is:

√(Σ(x-x̄)²/(n-1))

• x is each value
• x ̅ is the average of all values
• n is the number of values

Formula in cell D3:

Explaining the math formula

Here is the math formula Excel uses for calculating the standard deviation for a sample size:

x is each value
x ̅ is the average of all values
n is the number of values

Step 1 - Calculate the average

The arithmetic mean is calculated by adding all the values in a dataset and then dividing by the total number of values.

137+139+141+105+139+124+126+146+105+101 equals 1263.

1263/10 equals 126.3

Step 2 - Subtract the average from each value

137-126.3 = 10.7
139-126.3 = 12.7
141-126.3 = 14.7
105-126.3 = 21.3
139-126.3 = 12.7
124-126.3 = -2.3
126-126.3 = -0.3
146-126.3 = 19.7
105-126.3 = -21.3
101-126.3 = 25.3

Step 3 - Calculate the square

10.7² = 114.49
12.7² = 161.29
14.7² = 216.09
(-21.3)² = 453.69
12.7² = 161.29
(-2.3)² = 5.29
(-0.3)² = 0.09
19.7² = 388.09
(-21.3)² = 453.69
(-25.3)² = 640.09

Step 4 - Sum numbers

114.49+161.29+216.09+453.69+161.29+5.29+0.09+388.09+453.69+640.09 equals 2594.1

Step 5 - Divide sum by count minus 1

2594.1/(10-1)

becomes

2594.1/9

equals

288.233333333333

Step 6 - Calculate the square root

288.233333333333^(1/2) equals 16.9774360058677

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6. Sort rows by sample standard deviation

The image above shows a formula in cell B8 that sorts rows from cell range B3:N6 by the standard deviation of a sample from large to small. Column P contains these measures.

Here is the data:

	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
North	364	289	828	258	64	339	443	53	58	448	64	384
South	165	715	12	787	884	644	753	571	398	159	839	351
East	143	136	361	506	959	205	298	134	768	364	93	889
West	907	672	900	935	598	918	775	517	530	387	347	796

Formula in cell B8:

The formula returns the rows sorted based on the standard deviation calculated per row. Here is the output from that operation:

East	143	136	361	506	959	205	298	134	768	364	93	889
South	165	715	12	787	884	644	753	571	398	159	839	351
North	364	289	828	258	64	339	443	53	58	448	64	384
West	907	672	900	935	598	918	775	517	530	387	347	796

The Excel 365 formula spills values to cells below and to the right as far as needed, this is called spilling. A #SPILL! error occurs when at least one of the destination cells is non-empty.

Explaining formula

Step 1 - Calculate the standard deviation of a sample

STDEV.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,STDEV.S(a))

Step 3 - Calculate the standard deviation 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,STDEV.S(a)))

returns

{226.967091460548; 297.500445632465; 308.956699918035; 212.224937842904}

Step 4 - Sort rows based on the standard deviation 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,STDEV.S(a))),-1)

becomes

SORTBY(B3:N6,{226.967091460548; 297.500445632465; 308.956699918035; 212.224937842904},-1)

and returns

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Functions in 'Statistical' category

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

Excel function categories

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