Standard Deviation of Random Variables
Random variables are the numerical values that denote the possible outcome of the random experiment in the sample space. Calculating the standard deviation of the random variable tells us about the probability distribution of the random variable and the degree of the difference from the expected value.
We use X, Y, and Z as function to represent the random variables. The probability of the random variable is denoted as P(X) and the expected value is denoted by the μ symbol.
Then standard deviation of probability distribution is given using formula,
σ = √(∑ (xi – μ)2 × P(X)/n)
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Standard Deviation Formula Example
Example 1: Find the Standard Deviation of the following data,
xi | 5 | 12 | 15 |
---|---|---|---|
fi | 2 | 4 | 3 |
Solution:
First make the table as follows, so we can calculate the further values easily.
Xi | fi | Xi×fi | Xi-μ | (Xi-μ)2 | f×(Xi-μ)2 |
---|---|---|---|---|---|
5 | 2 | 10 | -6.375 | 40.64 | 81.28 |
12 | 3 | 36 | 0.625 | 0.39 | 1.17 |
15 | 3 | 45 | 3.625 | 13.14 | 39.42 |
Total | 8 | 91 |
|
| 121.87 |
Mean (μ) = ∑(fi xi)/∑(fi)
⇒ Mean (μ) = 91/8 = 11.375
σ = √(∑in fi(xi – μ)2/n)
⇒ σ = √[(121.87)/(8)]
⇒ σ = √(15.234)
⇒ σ = 3.90
Standard Derivation(σ) = 3.90
Example 2: Find Standard Deviation of following data table.
Class | Frequency |
---|---|
0-10 | 3 |
10-20 | 6 |
20-30 | 4 |
30-40 | 2 |
40-50 | 1 |
Solution:
Class | Xi | fi | f×Xi | Xi – μ | (Xi – μ)2 | f×(Xi – μ)2 |
---|---|---|---|---|---|---|
0-10 | 5 | 3 | 15 | -15 | 225 | 675 |
10-20 | 15 | 6 | 90 | -5 | 25 | 150 |
20-30 | 25 | 4 | 100 | 5 | 25 | 100 |
30-40 | 35 | 2 | 70 | 15 | 225 | 450 |
40-50 | 45 | 1 | 45 | 25 | 625 | 625 |
Total |
| 16 | 320 |
|
| 2000 |
Mean (μ) = ∑(fi xi)/∑(fi)
⇒ Mean (μ) = 320/16 = 20
σ = √(∑in fi(xi – μ)2/n)
⇒ σ = √[(2000)/(16)]
⇒ σ = √(125)
⇒ σ = 11.18
Standard Derivation(σ) = 11.18
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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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