Standard deviation is frequently used in statistics for different purposes in descriptive statistics, inferential statistics, and quality control. It has also wide uses in hypothesis testing, a measure of variability, normal distribution, and risk assessment.
In this article, we’ll learn the definition, types, and solved examples of the standard deviation.
In descriptive statistics, a measure that describes the variation and dispersion in the data set is said to be the standard deviation. It is a technique for measuring the data deviation from the expected value to measure the dispersion in a data set.
Types of standard deviation
There are two main types of standard deviation:
Here is a brief introduction to the types of standard deviation.
Sample Standard Deviation
In standard deviation, sample data is used to measure the variation in data observations and is said to be the sample standard deviation. The measure of variability is frequently used in this type of standard deviation. The sample standard deviation is denoted by “s”.
When the subset of a larger population is available, then this type of standard deviation is used. It is generally used when you want to estimate the standard deviation of the entire population based on the sample.
The formula for calculating the sample standard deviation is:
| $s=\sqrt{(\sum\left(x-\overline{x}\right)^{2}/\left(n-1\right)})$ |
-
s is the sample STD
-
x is each data point,
-
x̄ is the sample average (expected term)
-
Σ is the sum of all the data points
-
n is the total number in the sample subset of entire population.
|
Population Standard Deviation
In standard deviation, populationdata is used to measure the variation in data observations and is said to be the population standard deviation. The measure of variability is frequently used in this type of standard deviation. The populationstandard deviation is denoted by “σ”.
When entire observations of a larger population are available, then this type of standard deviation is used. It is generally used when you want to estimate the standard deviation of the entire population.
The formula for calculating the population standard deviation is:
| \[\sigma=\sqrt{(\sum\left(X-\mu\right)^{2}/N})\] |
-
σ is the population standard deviation
-
x is each data point
-
μ is the population mean
-
Σ is the sum of all the data points
-
N is the total number in the set of entire population.
|
A population standard deviation calculator is a useful resource to evaluate the problems of entire population data according to the formula.
Guidelines for working with a standard deviation
Here are some guidelines to keep in mind when working with standard deviation:
-
A lower standard deviation indicates that the data is clustered more closely around the mean.
-
The standard deviation cannot be negative since it represents a measure of distance from the mean.
-
Standard deviation can be influenced by extreme values.
What are the sample and population mean?
The sample mean and population mean are the average values of the sample and population data values that are helpful for the calculations of sample and population standard deviations.
|
Sample mean
|
Population mean
|
|
The sample mean is a statistical term that refers to the average of a subset of data taken from a larger population. It is also known as the "sample average."
|
Population mean is a statistical term that refers to the average of all the
values in a specific population. It is also known as the "population average."
|
|
The sample mean “x̄” is evaluated by adding up all the values “Σx” in the sample and dividing the sum “Σx” by the number of values in the sample.
|
Population mean “μ” is evaluated by adding up all the values
in the population and dividing the sum “Σx” by the number of values in the population.
|
|
It is denoted by “x̄” (x bar).
|
It is denoted by the symbol "μ" (mu).
|
|
The formula for calculating the sample mean is:
x̄ = (Σx) / n
|
The formula for calculating the sample mean is:
μ = (Σx) / n
|
|
Where
|
Where
|
Examples of sample and population standard deviation
Example-1: for population data
Find population standard deviation if:
42, 44, 48, 55, 60, 69, 71, 74, 75, 82
Solution
Step 1:First of all, calculate the population mean “μ”.
|
Given data
|
x = 42, 44, 48, 55, 60, 69, 71, 74, 75, 82
|
|
Sum
|
Σx = 42 + 44 + 48 + 55 + 60 + 69 + 71 + 74 + 75 + 82 = 620
|
|
population mean
|
μ = (Σx) ÷ n
μ = 620 ÷ 10 = 62
|
Step 2:Now take the difference of each population observation from the population mean and square them.
|
Data values
|
xi – μ
|
(xi - μ)2
|
|
42
|
42 – 62 = -20
|
(-20)2 = 400
|
|
44
|
44 – 62 = -18
|
(-18)2 = 324
|
|
48
|
48 – 62 = -14
|
(-14)2 = 196
|
|
55
|
55 – 62 = -7
|
(-7)2 = 49
|
|
60
|
60 – 62 = -2
|
(-2)2 = 4
|
|
69
|
69 – 62 = 7
|
(7)2 = 49
|
|
71
|
71 – 62 = 9
|
(9)2 = 81
|
|
74
|
74 – 62 = 12
|
(12)2 = 144
|
|
75
|
75 – 62 = 13
|
(13)2 = 169
|
|
82
|
82 – 62 = 20
|
(20)2 = 400
|
Step 3:Now add the values of the third column.
∑ (xi - μ)2 = 400 + 324 + 196 + 49 + 4 + 49 + 81 + 144 + 169 + 400
∑ (xi - μ)2 = 1816
Step 4:Now divide the above sum by n.
∑ (xi - μ)2 ÷ (n) = 1816 ÷ 10
∑ (xi - μ)2 ÷ (n) = 908 ÷ 5
∑ (xi - μ)2 ÷ (n) = 181.6
Step 5: Now take the square root.
√[∑ (xi - μ)2 ÷ (n)] = √181.6
√[∑ (xi - μ)2 ÷ (n)] = 13.48
Example-2: for sample data
Find sample standard deviation if:
12, 21, 8, 15, 20, 19, 31, 24, 15, 52, 36
Solution
Step 1:First of all, calculate the sample mean “x̄”.
|
Given data
|
x = 12, 21, 8, 15, 20, 19, 31, 24, 15, 52, 36
|
|
Sum
|
Σx = 12 + 21 + 8 + 15 + 20 + 19 + 31 + 24 + 15 + 52 + 36 = 253
|
|
Sample mean
|
x̄ = (Σx) ÷ n
x̄ = 253 ÷ 11 = 23
|
Step 2:Now take the difference of each sample observation from the sample mean and square them.
|
Data values
|
xi - x̄
|
(xi - x̄)2
|
|
12
|
12 – 23 = -11
|
(-11)2 = 121
|
|
21
|
21 – 23 = -2
|
(-2)2 = 4
|
|
8
|
8 – 23 = -15
|
(-15)2 = 225
|
|
15
|
15 – 23 = -8
|
(-8)2 = 64
|
|
20
|
20 – 23 = -3
|
(-3)2 = 9
|
|
19
|
19 – 23 = -4
|
(-4)2 = 16
|
|
31
|
31 – 23 = 8
|
(8)2 = 64
|
|
24
|
24 – 23 = 1
|
(1)2 = 1
|
|
15
|
15 – 23 = 8
|
(8)2 = 64
|
|
52
|
52 – 23 = 29
|
(29)2 = 841
|
|
36
|
36 – 23 = 13
|
(13)2 = 169
|
Step 3:Now add the values of the third column.
∑ (xi - x̄)2 = 121 + 4 + 225 + 64 + 9 + 16 + 64 + 1 + 64 + 841 + 169
∑ (xi - x̄)2 = 1578
Step 4:Now divide the above sum by n – 1.
∑ (Xi - x̄)2 ÷ (N – 1) = 1578 ÷ 11 – 1
∑ (Xi - x̄)2 ÷ (N – 1) = 1578 ÷ 10
∑ (Xi - x̄)2 ÷ (N – 1) = 789 ÷ 5 = 157.8
Step 5: Now take the square root.
√ [∑ (Xi - x̄)2 ÷ (N – 1)] = √157.8
√ [∑(Xi - x̄)2 ÷ (N – 1)] = 12.56
Conclusion
The standard deviation is explained with its definition, types, and solved examples in this post. Now you can grab all the basics of the standard deviation from this article for the concept to calculations.
