# Sample and Population Standard Deviation: Explained with Solved Examples

In statistics, the term standard deviation is widely used for various purposes. It is commonly used in different fields of education and daily life dealing such as weather forecast, finance, science, engineering, etc.

This statistical measurement is the square root of another statistical measurement known as a variance. In this post, we’ll learn sample and population standard deviations along with their formulas and solved examples.

## What is the standard deviation?

In statistics, the measure of the variability of a set of sample and population data is said to be the standard deviation. It is a well-known way to measure the spread of data such as whether it lies closer or far from the expected value.

The square root of the other statistical measure known as variance will give you the result of standard deviation. The term variance is the average of squared differences from the expected value of the set of population and sample data.

There are two types of data values in this measurement that is sample standard deviation and population standard deviation.

### Sample Standard Deviation

In statistics, the sample standard deviation is a type of STD that is used to measure the variability of a sample set of data. A sample is the smaller estimated values from a larger population of objects, individuals, or measurements.

For example, measuring the height of each student in the school is rather tough so we have to take the height of some students from the whole to measure estimate heights. The formula for measuring the sample standard deviation is:

 Formula Components s = √(Σ(xi– x̄)² / (n – 1)) s = the sample standard deviationΣ = the summation symbolxi = each data point in the samplex̄ = the sample meann = the total number of data points in the sample

### Population Standard Deviation

In statistics, the population standard deviation is a type of STD that is used to measure the variability of a population set of data. A population is the entire group of the larger population of objects, individuals, or measurements.

For example, measuring the height of each student in the school is the population estimation of heights. The formula for measuring the population standard deviation is:

 Formula Components σ = √(Σ(xi – μ)² / N) σ = the population standard deviationΣ = the summation symbolxi = each data point in the populationμ = the population meanN = the total number of data points in the population

Note

In the formula for sample standard deviation, we divide by (n-1) instead of n. This is because we are estimating the population standard deviation based on a sample, and dividing by n would underestimate the variability of the population.

### How to calculate the sample and population standard deviations?

Here are a few examples to calculate standard deviation of sample and population data values.

Example I: for population data

Calculate the given data values to measure the variability of the given entire experiments of an object.

2, 9, 12, 14, 16, 18, 22, 23, 25, 27, 30

Solution

Step 1: Find the sum of given experimental values and divide it by total numbers to calculate the expected value.

 Experiment values xi = 2, 9, 12, 14, 16, 18, 22, 23, 25, 27, 30 Addition of values Σ xi = 2 + 9 + 12 + 14 + 16 + 18 + 22 + 23 + 25 + 27 + 30Σ xi = 198 Population Average μ = (Σ xi ) ÷ nμ = 198 ÷ 11 = 18

Step 2: Now minus each experimental value from the expected value and take the square of subtracted results.

 Experimental values xi – μ (xi – μ)² 2 2 – 18 = -16 (-16)² = 256 9 9 – 18 = -9 (-9)² = 81 12 12 – 18 = -6 (-6)² = 36 14 14 – 18 = -4 (-4)² = 16 16 16 – 18 = -2 (-2)² = 4 18 18 – 18 = 0 (-0)² = 0 22 22 – 18 = 4 (4)² = 16 23 23 – 18 = 5 (5)² = 256 25 25 – 18 = 7 (7)² = 49 27 27 – 18 = 9 (9)² = 81 30 30 – 18 = 12 (12)² = 144

Step 3: Now take the sum of squared differences of data observations from the expected value.

Σ(xi– x̄)² = 256 + 81 + 36 + 16 + 4 + 0 + 16 + 25 + 49 + 81 + 144

Σ(xi- x̄)² = 708

Step 4: Now divide the above summed result by the total number of experimental values.

Σ(xi- x̄)² ÷ n = 708 ÷ 11

Σ(xi- x̄)² ÷ n = 64.364

Step 5: Now take the above result and find the square root of it.

√[Σ(xi- x̄)² ÷ (n)] = √64.364

√[Σ(xi- x̄)² ÷ (n)] = 8.023

Example-2: for sample data

Calculate the given data values to measure the variability of the given sample observations from the entire experiments of an object.

1, 9, 19, 23, 29, 31, 37, 39, 43, 49

Solution

Step 1: Find the sum of given experimental values and divide it by total numbers to calculate the expected value.

 Sample Observations xi = 1, 9, 19, 23, 29, 31, 37, 39, 43, 49 Sum of Sample Σ xi = 1 + 9 + 19 + 23 + 29 + 31 + 37 + 39 + 43 + 49Σ xi = 280 Sample Average x̄ = (Σ xi ) ÷ nx̄ = 280 ÷ 10 = 28

Step 2: Now minus each experimental value from the expected value and take the square of subtracted results.

 Experimental values xi – x̄ (xi – x̄)² 2 1 – 28 = -27 (-27)² = 729 9 9 – 28 = -19 (-19)² = 361 12 19 – 28 = -9 (-9)² = 81 14 23 – 28 = -5 (-5)² = 25 16 29 – 28 = 1 (1)² = 1 18 31 – 28 = 3 (3)² = 9 22 37 – 28 = 9 (9)² = 81 23 39 – 28 = 11 (11)² = 121 25 43 – 28 = 15 (15)² = 225 27 49 – 28 = 21 (21)² = 441

Step 3: Now take the sum of squared differences of data observations from the expected value.

∑ (xi  – x̄)² = 729 + 361 + 81 + 25 + 1 + 9 + 81 + 121 + 225 + 441

∑ (xi  – x̄)² = 2074

Step 4: Now divide the above sum by n – 1.

∑ (xi  – x̄)² ÷ (N – 1) = 2074 ÷ 10 – 1

∑ (xi  – x̄)² ÷ (N – 1) = 2074 ÷ 9

∑ (xi  – x̄)² ÷ (N – 1) = 230.44

Step 5: Now take the above result and find the square root of it.

√ [∑ (xi  – x̄)² ÷ (N – 1)] = √230.44

√ [∑ (xi  – x̄)² ÷ (N – 1)] = 15.18

### Conclusion

The term standard deviation is used to calculate the measure of the variability of sample and population data values. The formulas for calculating sample and population standard deviation are used to find the results easily and quickly.