Standard Deviation of Discrete Grouped Data
In grouped data first, we made a frequency table and then any further calculation was 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
Standard Deviation Formula Based on Discrete Frequency Distribution
For a given data set if it has n values (x1, x2, x3, …, xn) and the frequency corresponding to them is (f1, f2, f3, …, fn) then its standard deviation is calculated using the formula,
σ = √(∑in fi(xi – x̄)2/n)
where,
- n is Total Frequency (n = f1 + f2 + f3 +…+ fn )
- x̄ is Mean of Data
Example: Calculate the standard deviation for the given data
xi | fi |
---|---|
10 | 1 |
4 | 3 |
6 | 5 |
8 | 1 |
Solution:
Mean (x̄) = ∑(fi xi)/∑(fi)
⇒ Mean (μ) = (10×1 + 4×3 + 6×5 + 8×1)/(1+3+5+1)
⇒ Mean (μ) = 60/10 = 6
n = ∑(fi) = 1+3+5+1 = 10
xi | fi | fixi | (xi – x̄) | (xi – x̄)2 | fi(xi – 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 fi(xi – x̄)2/n)
⇒ σ = √[(16 + 12 + 0 +8)/10]
⇒ σ = √(3.6) = 1.897
Standard Derivation(σ) = 1.897
Standard Deviation of Discrete Data by Assumed Mean Method
In grouped data, 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,
di = xi – A
Now formula for standard deviation by assumed mean method is,
σ = √[(∑(fidi)2 /n) – (∑fidi/n)2]
where,
- ‘f‘ is Frequency of Data Value x
- ‘n‘ is Total Frequency [n = ∑(fi)]
Standard Deviation of Discrete Data by Step Deviation Method
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 also choose some arbitrary data value as the assumed mean (say A). Then we calculate the deviations of all data values (x1, x2, x3, …, xn), di = xi – A
In next step, we calculate the Step Deviations (d’) using
d’ = d/i
where ‘i‘ is Common Factor of all ‘d‘ values
Then, standard deviation formula is,
σ = √[(∑(fd’)2 /n) – (�’/n)2] × i
where ‘n‘ is Total Number of Data Values
Standard Deviation – Formula, Examples & How to Calculate
Standard Deviation is the measure of the dispersion of statistics. 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 to the mean value. There are different standard deviation formulas to calculate the standard deviation of a random variable.
In this article, we will learn about what is standard deviation, the standard deviation formulas, how to calculate standard deviation, and examples of standard deviation in detail.
Table of Content
- What is Standard Deviation?
- Standard Deviation Definition
- Standard Deviation Formula
- Formula for Calculating Standard Deviation
- How to Calculate Standard Deviation?
- What is Variance
- Difference between Variance and Deviation
- Varience Formula
- How to Calculate Variance?
- Standard Deviation of Ungrouped Data
- Standard Deviation of Discrete Grouped Data
- Standard Deviation of Continuous Grouped Data
- Standard Deviation of Probability Distribution
- Standard Deviation of Random Variables
- Standard Deviation Formula Example
- Standard Deviation Formula Excel
- Standard Deviation Formula Statistics
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