Calculation of Standard Deviation in Individual, Discrete & Continuous Series

In this article, we will discuss about Calculation of Standard Deviation in Individual, Discrete & Continuous Series and measures of dispersion in Statistics. How to calculate Standard deviation

Standard Deviation

Standard deviation Measures of Dispersion in Statistics is the measure of the dispersion of statistical data. The standard deviation formula is used to find the deviation of the data value from the mean value i.e. it is used to find the dispersion of all the values in a data set concerning the mean value. There are different standard deviation formulas to calculate the standard deviation of a random variable.

Standard deviation is a measure used in descriptive statistics to understand how the data points in a set are spread out from the average (mean) value. It indicates the extent of the data’s variation and shows how far individual data points deviate from the average

In this article, we will learn about what is standard deviation, the standard deviation formula, how to calculate standard deviation, and their examples in detail.

What is Standard Deviation?

Standard Deviation is defined as the degree of dispersion of the data point concerning the mean value of the data point. It tells us how the value of the data points varies concerning the mean value of the data point and it tells us about the variation of the data point in the sample of the data.

How to calculate Standard Deviation, measures of Dispersiopn, Statistics. The standard deviation of the given sample of the data set is also defined as the square root of the variance of the data set. The mean deviation of the n values (say x₁, x₂, x₃, …, x_n) is calculated by taking the sum of the squares of the difference of each value from the mean, i.e.

Mean Deviation = 1/n∑_iⁿ (x_i – x̄)²

The mean deviation is used to tell us about the scatter of the data. The lower degree of deviation tells us that the observations xi are close to the mean value and the depression is low, whereas the higher degree of deviation tells us that the observations xi are far from the mean value and the dispersion is high.

Calculation of Standard Deviation Formula

How to calculate Calculation of Standard deviation is used to measure the spread of the statistical data. It tells us about how the statistical data is spread out. Formula to Calculate Standard Deviation is used to find the deviation of all the data sets from its mean position. You may have questions that standard deviation how to calculate or how to calculate a standard deviation. Two standard deviation formulas are used to find the Standard Deviation of any given data set. They are,

Population Standard Deviation Formula
Sample Standard Deviation Formula

The formula for the standard deviation of sample data is,

$\bold{s = \sqrt{\frac{\sum_{i=1}^n (x_i - x̄)^2}{n-1}}}$

Notations for Standard Deviation,

s = Population Standard Deviation,

x_i= i^thobservation,

x̄ = Sample Mean, and

N = number of observations.

The standard deviation Formula of population data is,

$\bold{\sigma = \sqrt{\frac{\sum_{i=1}^N (x_i - \mu)^2}{N}}}$

Where,

σ = Population Standard Deviation,

x_i= i^thobservation,

μ = Population mean, and

N = number of observations.

It is evident to note that both formulas look the same and have only slide changes in their denominator. The denominator in the case of the sample is n-1 but in the case of the population is N. Initially the denominator in the sample standard deviation formula has “n” in its denominator but the result from this formula was not appropriate. So a correction was made and the n is replaced with n-1 this correction is called Bessel’s correction which in turn produced the most appropriate results.

The formula for Calculating Standard Deviation – Measures of Dispersion, Statistics | How to calculate Standard Deviation

The formula used for calculating the Standard Deviation is discussed in the image below, How to calculate Standard Deviation

How to Calculate Standard Deviation?

Generally, when we talk about standard deviation we talk about population standard deviation. The steps to calculate the standard deviation of a given set of values is,

Step 1: Calculate the mean of the observation using the formula (Mean = Sum of Observations/Number of Observations)

Step 2: Calculate the squared differences of the data values from the mean. (Data Value – Mean)²

Step 3: Calculate the average of the squared differences. (Variance = Sum of Squared Differences / Number of Observations)

Step 4: Calculate the square root of variance this gives the Standard Deviation. (Standard Deviation = √Variance)

Calculation of Standard Deviation in Individual Series

Standard Deviation of Ungrouped Data – Measures of Dispersion – Statistics

Calculation of standard deviation in Individual series or ungrouped data – Measures if Dispersion, Statistics, the standard deviation can be calculated using three methods that are,

Actual Mean Method
Assumed Mean Method
Step Deviation Method

How to Calculate Standard Deviation by Actual Mean Method

Calculation of Standard Deviation in Individual Series – Standard Deviation by actual mean method in individual series uses the basic mean formula to calculate the mean of the given data and using this mean value we find out the standard deviation of the given data values. We calculate the mean in this method with the formula,

μ = (Sum of Observations)/(Number of Observations)

and then the standard deviation is calculated using the standard deviation formula.

σ = √(∑_iⁿ (x_i – x̄)²/n)

Example: Find the Standard Deviation of the data set. X = {2, 3, 4, 5, 6}

Solution:

Given: n = 5, and observations x_i = {2, 3, 4, 5, 6}

We know,

Mean(μ) = (Sum of Observations)/(Number of Observations)

⇒ μ = (2 + 3 + 4 + 5 + 6)/ 5
⇒ μ = 4

σ² = ∑_iⁿ (x_i – x̄)²/n

⇒ σ² = 1/n[(2 – 4)²+ (3 – 4)²+ (4 – 4)²+ (5 – 4)²+ (6 – 4)²]
⇒ σ² = 10/5 = 2

Thus, σ = √(2) = 1.414

Standard Deviation by Assumed Mean Method

Calculation of Standard Deviation in Individual Series – For very large values of x finding the mean of the grouped data – Measures of Dispersion, Statistics is a tedious task so we assumed an arbitrary value (A) as the mean value and then calculated the standard deviation using the normal method in individual series. Suppose for the group of n data values ( x₁, x₂, x₃, …, x_n), the assumed mean is A then the deviation is,

d_i = x_i – A

Now, the assumed mean formula is,

σ = √(∑_iⁿ (d_i)²/n)

Standard Deviation by Step Deviation Method – Individual Series, Measures of Dispersion, Statistics

Calculation of Standard Deviation in Individual Series – We can also calculate the standard deviation of the grouped data or individual series using the step deviation method. As in the above method in this method also, we choose some arbitrary data value as the assumed mean (say A). Then we calculate the deviations of all data values (x₁, x₂, x₃, …, x_n), d_i = x_i – A

In the next step, we calculate the Step Deviations (d’) using

d’ = d/i

where ‘i‘ is a common factor of all ‘d’ values

Then, the standard deviation formula is,

σ = √[(∑(d’)² /n) – (∑d’/n)²] × i

where ‘n‘ is the total number of data values.

Calculation of Standard Deviation in Discrete Series – Measures of Dispersion

Calculation of standard deviation in Discrete Series In grouped data first, we made a frequency table and then any further calculation is made. For discrete grouped data, the standard deviation can also be calculated using three methods that are,

Actual Mean Method
Assumed Mean Method
Step Deviation Method

Calculation of Standard Deviation Formula in Discrete Series or Frequency Distribution

For a given data set of discrete series if it has n values (x₁, x₂, x₃, …, x_n) and the frequency corresponding to them is (f₁, f₂, f₃, …, f_n) then its standard deviation is calculated using the formula,

σ = √(∑_iⁿ f_i(x_i – x̄)²/n)

Where,

n is the total frequency (n = f₁ + f₂ + f₃ +…+ f_n ), and

x̄ is the mean of data

Example: Calculate the standard deviation for the given discrete series data

x_i	f_i
10	1
4	3
6	5
8	1

Solution:

Mean (x̄) = ∑(f_ix_i)/∑(f_i)

⇒ Mean (μ) = (10×1 + 4×3 + 6×5 + 8×1)/(1+3+5+1)
⇒ Mean (μ) = 60/10 = 6

n = ∑(f_i) = 1+3+5+1 = 10

x_i	f_i	f_ix_i	(x_i– x̄)	(x_i– x̄)²	f_i(x_i– x̄)²
10	1	10	4	16	16
4	3	12	-2	4	12
6	5	30	0	0	0
8	1	8	2	4	8

Now,

σ = √(∑_iⁿ f_i(x_i – x̄)²/n)

⇒ σ = √[(16 + 12 + 0 +8)/10]
⇒ σ = ????.6 = 1.897

Standard Derivation(σ) = 1.897

Standard Deviation of Discrete Data by Assumed Mean Method

In grouped data or discrete series also, if the values in the given data set are very large, then we assumed a random value (say A) as the mean of the data. Then the deviation of each value from the assumed mean is calculated as,

d_i = x_i – A

Now the formula for standard deviation by the assumed mean method in discrete series is,

σ = √[(∑(f_id_i)² /n) – (∑f_id_i/n)²]

Where,

‘f‘ is the frequency of data value x,

‘n‘ is the total frequency [n = ∑(f_i)].

Standard Deviation of Discrete Data by Step Deviation Method

Calculation of standard deviation in discrete series. We can also use the step deviation method to calculate the standard deviation of the discrete grouped data. As in the above method in this method also, we choose some arbitrary data value as the assumed mean (say A). Then we calculate the deviations of all data values (x₁, x₂, x₃, …, x_n), d_i = x_i – A

In the next step, we calculate the Step Deviations (d’) using

d’ = d/i

where ‘i‘ is a common factor of all ‘d‘ values

Then, the standard deviation formula is,

σ = √[(∑(fd’)² /n) – (∑fd’/n)²] × i

where ‘n‘ is the total number of data values.

Calculation of Standard Deviation in Continuous Series

Calculation of Standard Deviation in Continuous Series- For the continuous series or grouped data – measures of dispersion, statistics, we can easily calculate the standard deviation using the Discrete data formulas by replacing each class with its midpoint (as x_i) and then normally calculating the formulas.

The midpoint of each class is calculated using the formula,

x_i (Midpoint) = (Upper Bound + Lower Bound)/2

For example, calculate the standard deviation of the continuous grouped data as given in the table,

Class	0-10	10-20	20-30	30-40
Frequency(f _i )	2	4	2	2

Class

0-10

10-20

20-30

30-40

Frequency(f

)

Actual Mean Method in continuous series
Assumed Mean Method in continuous series
Step Deviation Method in continuous series

Calculation of Standard Deviation in Continuous Series – We can use any of the above methods to find or calculate of standard deviation in continuous series – measures of dispersion in statistics. Here we find the standard deviation using the actual mean method.

Solution to the above question is,

Class
5-15 15-25 25-35 35-45

x_i
10 20 30 40

Frequency(f_i)
2 4 2 2

Mean (x̄) = ∑(f_ix_i)/∑(f_i)

⇒ Mean (μ) = (10×2 + 20×4 + 30×2 + 40×2)/(2+4+2+2)
⇒ Mean (μ) = 240/10 = 24

n = ∑(f_i) = 2+4+2+2 = 10

x_i

f_i

f_ix_i

(x_i– x̄)

(x_i– x̄)²

f_i(x_i– x̄)²

10 2 20 14 196 392

20 4 80 -4 16 64

30 2 60 6 36 72

40 2 80 16 256 512

Now,

σ = √(∑_iⁿ f_i(x_i – x̄)²/n)

⇒ σ = √[(392 + 64 + 72 +512)/10]
⇒ σ = √104 = 10.198

Standard Derivation(σ) = 10.198

Class	5-15	15-25	25-35	35-45
x_i	10	20	30	40
Frequency(f_i)	2	4	2	2

x_i	f_i	f_ix_i	(x_i– x̄)	(x_i– x̄)²	f_i(x_i– x̄)²
10	2	20	14	196	392
20	4	80	-4	16	64
30	2	60	6	36	72
40	2	80	16	256	512

Similarly, other methods can also be used to find the standard deviation of the continuous series of grouped data.

Also Read:

Business Statistics eBook

Mock Tests and Test Series

Calculation of Mean

Median

Mode

A. Individual Series:

Deviation can be taken from Actual Mean and the following formula is used.

B. Discrete-Series:

Here also the deviations can be taken from Actual or Assumed Mean.

From Actual Mean:

S.D.=√∑fx²/N

Where x² is the square of deviations from actual mean, f denotes corresponding frequency; N = ∑ f

C. Continuous Series:

Here we take the deviations from Actual or Assumed Mean as desired from the Mid Point of Class-Intervals.

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Calculation of Standard Deviation in Individual, Discrete & Continuous Series | Statistics