How to calculate standard deviation? Explained with examples
In statistics, the term standard deviation is frequently used in various sub-branches. This term is usually used to calculate the variation in the data values. For example, the variation in the company expenses in the last 2 years can be evaluated by using standard deviation. Mainly, the process of calculating the standard deviation is almost the same as the variance, but the square root is taken of the out of variance to calculate the standard deviation. In the article, we will learn the definition, types, and how to calculate standard deviation along with examples.
What Is Standard Deviation?
In statistics, a technique that is used to measure the variability of sample and population data from the expected value of sample and population respectively is known as the standard deviation. Generally, the square root of the variance gives the output of the standard deviation.
The standard deviation is measured in single units while the variance is evaluated in squared units which is why the term standard deviation is more accurate than the variance. The relation of the data values is calculated with the help of variance and standard deviation.
The small output of the standard deviation represents that the data set of observations is closer to the mean. While the greater value of the standard deviation represents that the data values are far away from the mean and spread outward.
Types of standard deviation
On the basis of the nature of observation such as population and sample data values, the standard deviation is divided into two categories (types).
- Sample standard deviation
- Population Standard deviation
Sample Standard Deviation
The sample data observation is taken from the whole data set to measure the approximated values. The measure of dispersion or variability of the sample data observation from the sample mean is said to be the sample standard deviation. It is denoted by “s”. To calculate the sample standard deviation, follow the below steps:
- First, add the given set of observations and divide it by the total number of observations to get the sample mean (x̄).
- Calculate the difference between each observation from the sample mean. It is known as the deviation.
- After that take the square of deviations to make them positive.
- Find the sum of squared deviations.
- Divide the sum of squared deviation by the degree of freedom (n – 1).
- Take the square root of the result of the quotient.
The general formula of the sample standard deviation is:
s = √ [∑ (xi – x̄)2 / n – 1]
Population Standard Deviation
The population data observation is taken for the whole data set. The measure of dispersion or variability of the population data observation from the population means is said to be the population standard deviation. It is denoted by “σ”. To calculate the population standard deviation, follow the below steps:
- First, add the given set of observations and divide it by the total number of observations to get the population mean (µ).
- Calculate the difference between each observation from the population mean. It is known as the deviation.
- After that take the square of deviations to make them positive.
- Find the sum of squared deviations.
- Divide the sum of the squared deviation by the total number of observations.
- Take the square root of the result of the quotient.
The general formula of the population standard deviation is:
σ = √ [∑ (xi – µ)2 / n]
Standard Deviation in Calculator
Statistics problems of sample and population standard deviation can be solved with the help of the mean and standard deviation calculator according to the above steps. You can achieve the same results by calculating the standard deviation in SPSS and Excel data tools.
How To Calculate Standard Deviation
To calculate the standard deviation, the formulas of sample and population standard deviation play a vital role. Let us take a few examples of standard deviation to learn how to calculate it.
Sample standard deviation formula
Calculate the standard deviation of the given sample data:
3, 7, 6, 14, 18, 12, 20, 24, 36, 44, 47
Solution:
Step 1: First, add the given observations and divide them by the total number of terms to calculate the sample mean.
Sum = 3 + 7 + 6 + 14+ 18 + 12 + 20 + 24 + 36 + 44 + 47
Sum = 231
Total number of observation = N = 11
Sample Mean = x̄ = 231/11
Sample Mean = x̄ = 21
Step 2: Now find the difference between the observations and mean. After that evaluate the square of deviations.
Data | xi – x̄ | (xi – x̄)2 |
3 | 3 – 21 = -18 | (-18)2= 324 |
7 | 7 – 21 = -14 | (-14)2= 196 |
6 | 6 – 21 = -15 | (-15)2= 225 |
14 | 14 – 21 = -7 | (-7)2= 49 |
18 | 18 – 21 = -3 | (-3)2= 9 |
12 | 12 – 21 = -9 | (-9)2= 81 |
20 | 20 – 21 = -1 | (-1)2= 1 |
24 | 24 – 21 = 3 | (3)2= 9 |
36 | 36 – 21 = 15 | (15)2= 225 |
44 | 44 – 21 = 23 | (23)2= 529 |
47 | 47 – 21 = 26 | (26)2= 676 |
Step 3: Evaluate the sum of the deviations.
∑ (xi – x̄)2 = 324 + 196 +225 + 49 + 9 + 81 + 1 + 9 + 225 + 529 + 676
∑ (xi – x̄)2 = 2324
Step 4: Divide the sum of squared deviations by the degree of freedom (n-1).
∑ (xi – x̄)2 / n – 1 = 2324/ 11 – 1
∑ (xi – x̄)2 / n – 1 = 2324/ 10
∑ (xi – x̄)2 / n – 1 = 232.4
Step 5: Take the square root of the above expression to calculate the sample standard deviation.
√[∑ (xi – x̄)2 / n – 1] = √232.4
√[∑ (xi – x̄)2 / n – 1] =15.25
Population Standard Deviation Formula
Calculate the standard deviation of the given population data below:
5, 3, 8, 1, 12, 14, 13, 16
Solution:
Step 1: First, add the given observations and divide them by the total number of terms to calculate the population mean.
Sum = 5 + 3 + 8 + 1 + 12 + 14 + 13 + 16
Sum = 72
Total number of observation = n = 8
Population Mean = µ = 72/8
Population Mean = µ = 9
Step 2: Now find the difference between the observations and mean. After that evaluate the square of deviations.
Data values | xi – µ | (xi – µ)2 |
5 | 5 – 9 = -4 | (-4)2 = 16 |
3 | 3 – 9 = -6 | (-6)2 = 36 |
8 | 8 – 9 = -1 | (-1)2 = 1 |
1 | 1 – 9 = -8 | (-8)2 = 64 |
12 | 12 – 9 = 3 | (3)2 = 9 |
14 | 14 – 9 = 5 | (5)2 = 25 |
13 | 13 – 9 = 4 | (4)2 = 16 |
16 | 16 – 9 = 7 | (7)2 = 49 |
Step 3: Evaluate the sum of the deviations.
∑ (xi – µ)2 = 16 + 36 + 1 + 64 + 9 + 25 + 16 + 49
∑ (xi – µ)2 = 216
Step 4: Divide the sum of squared deviations by total number of observations.
∑ (xi – µ)2 / N = 216 / 8
∑ (xi – µ)2 / N = 54 / 2
∑ (xi – µ)2 / N = 27
Step 5: Take the square root of the above expression to calculate the population standard deviation.
√[∑ (xi – µ)2 / N] = √27
√[∑ (xi – µ)2 / N] = 5.196
Summary
In conclusion, you can grab all the basics of calculating the standard deviation from this post as we have discussed the definitions, types, formulas, and given examples with the solution. As you have witnessed, this topic is not a tough one, it just requires more effort. Similar to mathematics, statistics has no shortcuts and requires constant practice. However, if you feel that learning the formulae is too much, you can always pay someone to do your statistics homework for you.