Standard deviation: Introduction

In descriptive statistics, the term standard deviation is used on a wider scale to measure the spread of data values from the expected value. It is used in hypothesis testing and many other well-known branches of statistics.

It is the square root of the variance such as taking the square root of the result of the variance will give you the output of the standard deviation. In this lesson, we will learn all the basics of standard deviation along with solved examples.

What is the standard deviation?

In descriptive statistics, the standard deviation (SD) is a technique that is used to measure the spread of sample or population data values from the sample or population mean respectively. In general, the standard deviation is the square root of the output of variance.

It is more accurate than the variance as in this technique the units of measurement are not in squared form. It is mostly used to measure the dispersed of the data values from the expected values to get the relation between them.

If the value of the standard deviation is small, then it indicates that the data values are collected (clustered) near the mean. While the larger value of the standard deviation indicates that the data are scattered away from the mean.

Kinds of the standard deviation

On the basis of the nature of the data values, there are two subtypes of the standard deviation.

• Sample standard deviation
• Population standard deviation

Let us briefly describe the subtypes of the standard deviation along with examples.

1.   Population standard deviation

The term population is referred to the whole set of data values as the population is taken for all the objects or things such as the number of girls in a city. The measure of the spread of data values from the expected value of the population is known as the population standard deviation.

It can be measured by taking the difference between the data values from the population mean and then taking the square of the mean to make them positive as the standard deviation is always positive. After that add all the squared terms and divide the sum of squares by the total number of population data.

The general expression of this sub-branch of the standard deviation is:

σ = √ [∑ (z_i – μ)²/N]

In the above population standard deviation formula, the symbol “σ” denotes the population standard deviation and is read as sigma, “N” is the total number of population data values, z_i is the given data values of the population set, the “μ” is referred to the population mean of the given set of population data values, and “z_i – μ” is the mean deviation of all the observations from the mean.

Example

Measure the spread of the given data values to calculate the standard deviation

1, 5, 9, 14, 18, 20, 24

Solution

Step 1: First of all, calculate the population mean of the given data values. 

Sum = 1 + 5 + 9 + 14 + 18 + 20 + 24

Sum = 91

Total number of observation = N = 7

Population Mean = µ = 91/7

Population Mean = µ = 13

Step 2: Now calculate the difference of data values from the mean and take the square of the differences to make them positive.

Step 3: Now add all the squared differences.

∑ (z_i – µ)² = 144 + 64 + 16 + 1 + 25 + 49 + 121

∑ (z_i – µ)² = 420

Step 4: Now take the quotient of the sum of squared differences and the total number of observations. This will give you the result of the variance.

∑ (z_i – µ)² / N = 420 / 7

∑ (z_i – µ)² / N = 60 / 1

∑ (z_i – µ)² / N = 60

Step 5: Now take the above result and apply the square root to it to calculate the population standard deviation.

√[∑ (z_i – µ)² / N] = √60

√[∑ (z_i – µ)² / N] = 7.75

A standard deviation calculator can be used to ease up such a lengthy calculations for finding the standard deviation of sample or population set of data values.

2.   Sample standard deviation

The term sample is referred to some values from the whole set of data values as the population is taken for all the objects or things so to ease up the calculation we take the sample set of data values to estimate the numbers.

The measure of the spread of data values from the expected value of the sample is known as the sample standard deviation. It can be measured by taking the difference between the data values from the sample mean and then taking the square of the mean to make them positive as the standard deviation is always positive.

After that add all the squared terms and divide the sum of squares by the total number of sample data minus one.

The general expression of this sub-branch of the standard deviation is:

s = √ [∑ (z_i – z̄)²/N – 1]

In the above sample standard deviation formula, the symbol “s” denotes the sample standard deviation and is read as sigma, “N” is the total number of sample data values, z_i is the given data values of the sample set, the “z̄” is referred to the sample mean of the given set of sample data values, and “z_i – z̄” is the mean deviation of all the observations from the mean.

Example

Measure the spread of the given data values to calculate the standard deviation

6, 12, 16, 15, 19, 23, 27, 28, 30, 34

Solution

Step 1: First of all, calculate the sample mean of the given data values.  

Sum = 6 + 12 + 16 + 15 + 19 + 23 + 27 + 28 + 30 + 34

Sum = 210

Total number of observation = N = 10

Sample Mean = z̄ = 210/10

Sample Mean = z̄ = 21

Step 2: Now calculate the difference of data values from the mean and take the square of the differences to make them positive.

Step 3: Now add all the squared differences.

∑ (z_i – z̄)² = 225 + 81 + 25 + 36 + 4 + 4 + 36 + 49 + 81 + 169

∑ (z_i – z̄)² = 710

Step 4: Now take the quotient of the sum of squared differences and the total number of observations decreased by one. This will give you the result of the variance.

∑ (z_i – z̄)² / N – 1 = 710 / 10 – 1

∑ (z_i – z̄)² / N – 1 = 710 / 9

∑ (z_i – z̄)² / N – 1 = 78.89

Step 5: Now take the above result and apply the square root to it to calculate the sample standard deviation.

√[∑ (z_i – z̄)² / N – 1] = √78.89

√[∑ (z_i – z̄)² / N – 1] = 8.88

Wrap up

Now you can get all the basics of the standard deviation from this post. In this post, we have covered all the basics that are necessary to solve the problems of the standard deviation with definitions, types, formulas, and solved examples.

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