Standard deviation is a statistical term that measures how much
the observed value is far from their means. If the difference between the
actual value and means increases then the standard deviation also increases. First,
we should know the difference between grouped data and ungrouped data to move the
standard deviation.
Grouped data is in the form of class interval and have
frequency. Like, 05, 610, 1115, and so on. However, ungrouped data refers to
raw information that is collected randomly and not sorted into classified. For
example 14,16,89,35.
We will discuss standard deviation formulas for both grouped
and ungrouped data. Means, Variance, and standard deviation formulas are quite
related. We will cover all terms that relate to standard deviation.
What is Standard deviation (SD) in statistics?
Standard deviation refers to how much each observed value
deviates from its means. If the observed value deviates from its means then the
standard deviation is less. It can be defined as the square root of the means
of the square of the deviation value from the means. In other words, we can say
the under root of the variance is the standard
deviation.
Different Standard deviation formulas
Let’s discuss the different formulas of standard deviation
Population standard
deviation (Ïƒ) and sample standard deviation (s) formulas
The population is
the entire group or event in which we are interested. However, a sample is any
subgroup of the population. Population standard deviation is denoted by the
Greek letter sigma (lower case) “Ïƒ”
and “S” indicates sample standard deviation.
Population standard deviation 
sample standard deviation 
Ïƒ = Sqrt [Ã¥ (x – Î¼)^{ 2} / N] And, Î¼ = Sqrt [Ã¥ x / N] Here, Î¼ = population means x = observed values N = total number of observed values 
S =
Sqrt [Ã¥ (x
– x̄)^{
2} / (n – 1)] And, x̄ = Sqrt [Ã¥ x / n] Here, x̄ = sample means x = observed values n = total number of observed values 
A standard deviation calculator (https://www.standarddeviationcalculator.io/)
is a help source to obtain the standard deviation of the given set of data
according to the formula of sample and population STD.
SD for ungrouped data
·
Direct method: S = Sqrt [Ã¥ (x – x̄)^{
2} / n]
·
Shortcut method :S = Sqrt [(Ã¥ x^{ 2} / n)  (Ã¥ x^{ }/ n)^{ 2}]
SD for grouped data
There are two conditions. If only class intervals are given
but their frequency is not known then we find the midpoints (x) by adding lowerclass
interval and upperclass interval and then dividing it by 2. We solve it by
ungrouped method.
If we have class intervals as well as their frequency. Then
we have
S.D = Sqrt [Ã¥ f(x – x̄)^{ 2} / Ã¥f]
Standard deviation properties
Some important properties of standard deviation are given
below,
·
Standard deviation of identical observed values
is zero.
·
Standard deviation is independent of his region
i.e. S.D (x ±
a) = S.D (x)
·
If a is any constant then S.D (ax) = a S.D (x)
and S.D (x / a) = 1 / a S.D(x)
·
If x and y are independent variables then S.D (x
±
y) = S.D (x) + S.D (y)
Method of finding the standard deviation by a direct method
Let’s learn how we can find standard deviation by a direct
method.
·
First, we calculate arithmetic means (average).
·
Find (x – x̄) for each
given observation, then calculate the square of these values.
·
Find the Sum of all (x – x̄)^{2}
·
To get the standard deviation, divide Ã¥ (x – x̄)^{ 2} by the total number of observations.
Solved Examples of Standard deviation
Example1.
Calculate the standard deviation for the following
observation
2, 4, 6, 8
First, we will find arithmetic means (x̄)
Arithmetic means = sum of all observations / total number of
observation
x̄ = 2 + 4 + 6 + 8 / 4 = 20
x 
(x – x̄) 
(x – x̄)^{2} 
2 
2 – 20 = 18 
(18)^{2} = 324 
4 
4 – 20 = 16 
(16)^{2 }= 256 
6 
6 – 20 = 14 
(14)^{2 }= 196 
8 
8 – 20 = 12 
(12)^{2 }= 144 


Ã¥(x – x̄)^{2} = 920 
Ã¥(x – x̄)^{ 2} = 1738.996
Put it in the standard deviation formula i.e. S = Sqrt [Ã¥ (x – x̄)^{ 2} / n], we have
S = Sqrt [920 / 4] = 230
S = 230
So, the standard deviation of 2, 4, 6, and 8 is 230
Example 2
Find the standard deviation for the following values,
65, 46, 56, 75, 54, 64, 79, 95
Solution
Arithmetic Means = x̄ = 66.75
x 
(x – x̄) 
(x – x̄)^{2} 
65 
65 – 66.75 = – 1.75 
3.062 
46 
46 – 66.75 = – 20.75 
430.562 
56 
56 – 66.75 = – 10.75 
115.562 
75 
75 – 66.75 = 8.25 
68.062 
54 
54 – 66.75 = – 12.75 
162.562 
64 
64 – 66.75 = – 2.75 
7.562 
79 
79 – 66.75 = 12.75 
162.562 
95 
95 – 66.75 = 28.25 
789.062 


Ã¥(x – x̄)^{2} = 1738.996 
As we know that
S.D = Sqrt [Ã¥
(x – x̄)^{ 2} / n]
Now put the values, we have
S.D = Sqrt [1738.996 / 8] = 217.3745
S.D = Sqrt [217.3745] = 14.743
The standard deviation of the given observation is 14.743.
Conclusion
In this article, we have discussed the standard deviation in
detail. Then we described the definition of standard deviation. Then we
discussed the formula of population and sample standard deviation, also
discussed the formula for grouped and ungrouped data. Some properties of
standard deviation covered in this article.
After this, we learned how to calculate standard deviation
by the direct method. Then we solved the example of standard definition in a
good manner. After reading this article, you can easily solve any question
related to standard deviation.
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