Standard Deviation in Discrete Series |

Standard deviation is a crucial statistical measure that indicates the dispersion or variability of a set of data points around the mean (average). In a Discrete Series, each observation is paired with a corresponding frequency, which shows how many times that observation occurs.

Formula for Standard Deviation (σ) in a Discrete Series:
𝜎
=
∑
𝑓
𝑖
(
𝑥
𝑖
−
𝑥
‾
)
2
∑
𝑓
𝑖
σ=
∑f
i
​

∑f
i
​
(x
i
​
−
x
)
2

​

​


Where:

𝑥
𝑖
x
i
​
= Individual values
𝑓
𝑖
f
i
​
= Corresponding frequencies
𝑥
‾
x
= Mean of the series
𝜎
σ = Standard Deviation
Steps to Calculate Standard Deviation:
Calculate the Mean (
𝑥
‾
x
):

𝑥
‾
=
∑
𝑓
𝑖
𝑥
𝑖
∑
𝑓
𝑖
x
=
∑f
i
​

∑f
i
​
x
i
​

​

Multiply each value (
𝑥
𝑖
x
i
​
) by its frequency (
𝑓
𝑖
f
i
​
).
Sum these products.
Divide by the total frequency (
∑
𝑓
𝑖
∑f
i
​
) to get the mean.
Find the Deviations (
𝑥
𝑖
−
𝑥
‾
x
i
​
−
x
):

Subtract the mean from each value (
𝑥
𝑖
x
i
​
).
Square the Deviations and Multiply by Corresponding Frequencies:

𝑓
𝑖
(
𝑥
𝑖
−
𝑥
‾
)
2
f
i
​
(x
i
​
−
x
)
2

Square the deviations calculated in step 2.
Multiply each squared deviation by its corresponding frequency.
Sum the Squared Deviations:

∑
𝑓
𝑖
(
𝑥
𝑖
−
𝑥
‾
)
2
∑f
i
​
(x
i
​
−
x
)
2

Add up all the squared deviations multiplied by their frequencies.
Calculate the Standard Deviation:

𝜎
=
∑
𝑓
𝑖
(
𝑥
𝑖
−
𝑥
‾
)
2
∑
𝑓
𝑖
σ=
∑f
i
​

∑f
i
​
(x
i
​
−
x
)
2

​

​

Divide the sum of squared deviations by the total frequency.
Take the square root of the result to get the standard deviation.
Example:
Suppose you have the following data:

𝑥
𝑖
x
i
​
(Value)
𝑓
𝑖
f
i
​
(Frequency)
10 3
20 5
30 2
Calculate the Mean (
𝑥
‾
x
):

𝑥
‾
=
(
10
×
3
)
+
(
20
×
5
)
+
(
30
×
2
)
3
+
5
+
2
=
30
+
100
+
60
10
=
19
x
=
3+5+2
(10×3)+(20×5)+(30×2)
​
=
10
30+100+60
​
=19
Find Deviations and Squared Deviations:

𝑥
𝑖
x
i
​

𝑓
𝑖
f
i
​

𝑥
𝑖
−
𝑥
‾
x
i
​
−
x

(
𝑥
𝑖
−
𝑥
‾
)
2
(x
i
​
−
x
)
2

𝑓
𝑖
(
𝑥
𝑖
−
𝑥
‾
)
2
f
i
​
(x
i
​
−
x
)
2

10 3 -9 81 243
20 5 1 1 5
30 2 11 121 242
∑
𝑓
𝑖
(
𝑥
𝑖
−
𝑥
‾
)
2
=
243
+
5
+
242
=
490
∑f
i
​
(x
i
​
−
x
)
2
=243+5+242=490
Calculate Standard Deviation (
𝜎
σ):

𝜎
=
490
10
=
49
=
7
σ=
10
490
​

​
=
49
​
=7
So, the standard deviation for this discrete series is 7.

Conclusion:
Standard deviation gives insight into how spread out the values in a discrete series are from the mean. A lower standard deviation means the data points are closer to the mean, while a higher standard deviation indicates more variability.

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