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How to Find Standard Deviation

The standard deviation is the most important and widely used measure of studying variation (dispersion). It shows the variation in data. The calculation of the standard deviation is a bit complicated. The risk of making a mistake is high, so we need high attention and accurate calculation. In this section, we will learn how to find the standard deviation.

Standard Deviation Definitions

The standard deviation (SD) is a quantification that measures the distribution (dispersion) of the data set relative to its mean. It is calculated as the square root of the variance. It is denoted by the lower Greek letter σ (sigma). If the deviation is greater, the dispersion will be greater, and if the deviation is lesser greater the uniformity.

Some other definitions are:

• The standard deviation is the measure of the variations of all values from the mean.
• Standard deviation is the square root of the sum of squared deviation from the mean divided by the number of observations.
• It is the square root of the variance.

Variance

It defines how much a random variable differs from its expected value. It is the average of the squares of the differences between expected and individual value. It can never have a negative value. It is denoted by σ². The formula of variance is:

How to Find Standard Deviation

It is calculated as the square root of variance by determining the variations between each data point relative to the mean. The higher the standard deviation, the higher the variance between each data set, and the mean.

The formula of Standard Deviation

There are two formulas to calculate the standard deviation. Both formulas measure the variations. But there is a difference between them.

• Population Standard Deviation
• Sample Standard Deviation

Population Standard Deviation

It is a parameter that calculates a fixed value from every individual in the population. The formula of population standard deviation is:

Where:

σ: Population standard deviation.

x_i: Each element in the data

set. Where i = 1, 2, 3, …., N.

μ: Mean of all elements in the data set.

N: The number of elements.

Sample Standard Deviation

It is a statistic. In this standard deviation, only some individuals from the population are taken for the calculation. It has greater variability because it depends on the sample. Hence, the standard deviation of the sample is greater than the population standard deviation.

The formula of sample standard deviation is:

Where:

s: Sample standard deviation.

x_i: Each element in the data set. Where i = 1, 2, 3, …., N.

x: Mean of all elements in the data set.

N: The number of elements.

Now, we will see how these standard deviations are different from each other. Consider the sample and population standard deviation formula; we see that both the formulas are nearly identical.

Step 1: First, calculate the mean. Sum up all the values and divide by the number of elements.

Step 2: Calculate the deviation from the mean. To achieve the same, subtract the mean from each value.

Step 3: Square the deviations.

Step 4: Square the deviations and add them.

Step 5: Divide the squared deviation by the number of observations. This step has a major difference between population standard deviation and sample standard deviation.

• While using the population standard deviation, divide the sum of the squared deviation by N (number of elements or observations).
• While using sample standard deviation, divide the sum of the squared deviation by N-1 (one less than the number of elements or observations).

Step 6: Find the square root of the quotient that we get in the above step.

The values of population and sample standard deviation depend on N. The larger the value of N, the greater the population, and sample standard deviation.

Properties of Standard Deviation

• The value of standard deviation is never negative.
• The low deviation indicated that the data point tends to very closer to the mean.
• High deviation indicates that the data point is spread out over a large range of values.
• If we add a constant to all the data sets, it does not affect the standard deviation.
• If we multiply a constant to all the data set, it affects the standard deviation.
• The standard deviation can be zero if and only if all the observations have the same value.

Uses of Standard Deviation

• It is widely used in biological studies, statistics, and financial field.
• It is used in fitting a normal curve to a frequency distribution.
• It is used to measure of dispersion.
• It is also used in the finance field to calculate financial risk.

Methods of Standard Deviations

Direct Method

We can also find the standard deviation by using the direct method. It is used when the deviation is taken from the actual mean. The formula for the direct method is:

Where:

d=(x_i–x)

σ: Standard deviation

x_i: Each element in the data set. Where i = 1, 2, 3, …., N.

x: Mean of all elements in the data set.

N: The number of elements.

Assumed Mean Method

In this method, we do not calculate the actual mean. Instead of this, we choose a random value to calculate the deviation. The assumed value must lie around the middle value. It is also known as the shortcut method. The formula for the assumed mean method is:

Where,

f: Corresponding frequency

d=x-A (A is assumed mean)

N: The number of elements in the data set.

Step Deviation Method

It is an extended form of the shortcut method. It simplifies the calculation. The formula for the assumed mean method is:

Where,

f: Corresponding frequency

d= (A is assumed mean)

N: The number of elements in the data set.

i: Common class interval

Types of Distributions

Before moving to the examples, we must know about the three types of distribution.

• Individual Series: Individual series is a single column observation. For example:

Marks (x)

55

34

78

58

90

67

81

• Discrete Series: In discrete series, there are two columns. The first column consists of the observations, and the second column consists of the frequencies. For example:

Marks (x)	65	86	58	45	88	90	35
No. of Students (f)	5	7	12	8	4	2	1

• Frequency Distribution: Frequency distribution also has two columns. The first column consists of the observations, and the second column consists of the frequencies. The observations are further classified into the intervals called classes. For example:

Marks (x)	20-30	30-40	40-50	50-60	60-70	70-80	80-90
No. of Students (f)	6	8	14	7	3	9	2

Standard Deviation Formulas
Distribution	Direct Method	Assumed Mean or Short-cut Method	Step Deviation Method
Individual Series			–
Discrete Series			–
Frequency Distribution	–	–

Example of Individual Series

Example: Find the standard deviation of the following data using direct and assumed mean method.

Marks

25

34

21

28

60

33

72

55

Solution:

Using the Direct Method

First, we will calculate the mean.

Now, we will calculate the variance (σ²).

The formula for variance is:

Marks (x)	d=(xi–x)	d2=(xi–x)2
25	-16	256
34	-7	49
21	-20	400
28	-13	169
60	19	361
33	-8	64
72	31	961
55	15	225
	∑(xi–x)=1	∑(xi–x)2= 2485

Putting the values in the variance formula, we get:

The formula for standard deviation is: σ =√σ²

σ =√310.625=17.624
σ =17.624

Using Assumed Mean or Short-cut Method

We know the formula of the assumed mean method for individual series:

In the above formula, d=x-A. Where A is assumed mean. So, let’s assume A = 38.

Marks (x)	d=(xi-A)	d2=(xi-A)2
25	-13	169
34	-4	16
21	-17	289
28	-10	100
60	22	484
33	-6	36
72	34	1156
55	17	289
	∑(xi-A)=23	∑(xi-A)2= 2539

Putting the values in the above formula, we get:

Example of Discrete Series

Example: Find the standard deviation of the data given below using the direct and shortcut method.

Marks (x)	3.5	4.5	5.5	6.5	7.5	8.5	9.5
No. of Students (f)	3	7	22	60	85	32	8

Solution:

Using the Direct Method

First, we will calculate the mean.

We know the formula of a direct method for discrete series:

Marks (x)	f	d=(xi–x)	d2=(xi–x)2	fd	fd2
3.5	3	-3	9	-9	27
4.5	7	-2	4	-14	28
5.5	22	-1	1	-22	22
6.5	60	0	0	0	0
7.5	85	1	1	85	85
8.5	32	2	4	64	128
9.5	8	3	9	24	72
	∑f=217				∑fd2=362

Putting the values in the formula, we get:

Using the Short-cut Method

We know the formula of the shortcut method for discrete series:

In the above formula, d=x-A. Where A is assumed mean. So, let’s assume A = 6.5.

Marks (x)	f	d=(xi-A)	d2=(xi-A)2	fd	fd2
3.5	3	-3	9	-9	27
4.5	7	-2	4	-14	28
5.5	22	-1	1	-22	22
6.5	60	0	0	0	0
7.5	85	1	1	85	85
8.5	32	2	4	64	128
9.5	8	3	9	24	72
	∑f=217			∑fd=128	∑fd2=362

Putting the values in the formula, we get:

Hence, the standard deviation is 1.148.

Example of Frequency Distribution (Grouped Data or Continuous Series)

Example: Calculate the standard deviation of the data given below using the direct and shortcut method.

Marks (x)	0-10	10-20	20-30	30-40	40-50	50-60	60-70	70-80
No. of Students (f)	15	15	23	22	25	10	5	10

Solution:

Using Step Deviation Method

We know the formula of step deviation method for continuous series:

In the above formula, . Where A is assumed mean. So, first, we will calculate the mean (m). In the following table, we have calculated the mean of each class interval. Among them, we have assumed a mean that is 35.

Marks (x)	f			d2	fd	fd2
0-10	15	5	-3	9	-45	135
10-20	15	15	-2	4	-30	60
20-30	23	25	-1	1	-23	23
30-40	22	35 (A)	0	0	0	0
40-50	25	45	1	1	25	25
50-60	10	55	2	4	20	40
60-70	5	65	3	9	15	45
70-80	10	75	4	16	40	160
	∑f=N=125				∑fd=2	∑fd2=488

Putting the values in the formula, we get:

Example of Population Standard Deviation

Example: Find the standard deviation using the population standard deviation.

12, 2, 45, 23, 55, 8, 11, 19, 57, 3

Solution:

In the above question, the marks of ten students are given. The question says that apply the sample standard deviation. In this case, we will not take all student’s marks for calculation. We will take a few student’s marks for calculation as a sample.

We have taken only six marks for calculation are as follows:

12, 45, 23, 11, 19, 3

We know the formula of sample standard deviation:

Now, we will find the values used in the formula.

Step 1: Calculate the sample mean (x).

Step 2: For each data element, subtract the mean and square the result.

x	(xi–x)	(xi–x)2
12	-7	49
45	26	676
23	4	16
11	-7	49
19	0	0
3	-16	256
		∑(xi–x)2=1046

Step 3: Divide the ∑(x_i–x)² by the N-1. Here, a total of 6 elements are there, so dividing the sum by 6-1=5, we get:

Step 4: Take the square root of the above result.

s =√209.2=14.46

Hence, the sample standard deviation is 14.46.

Example of Sample Standard Deviation

Example: Find the standard deviation using the population standard deviation.

12, 2, 45, 23, 55, 8, 11, 19, 57, 3

Solution:

In the above question, the marks of ten students are given. The question says that apply the population standard deviation. In this case, we will take all student’s marks for calculation.

We know the formula of sample standard deviation:

Now, we will find the values used in the formula.

Step 1: Calculate the population mean (μ).

Step 2: For each data element, subtract the mean and square the result.

x	(xi-μ)	(xi-μ)2
12	-12	144
2	-22	484
45	21	441
23	-1	1
55	31	961
8	-16	256
11	-13	169
19	-5	25
57	33	1089
3	-21	441
		∑(xi-μ)2=4011

Step 3: Divide the ∑(x_i-μ)² by N. Here, a total of 10 elements are there, so dividing the sum by 10, we get:

Step 4: Take the square root of the above result.

σ =√401=20.02=20

Hence, the population standard deviation is 20.

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