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Standard deviation formula is generally used to calculate the values of a given data set that is scattered. In general, the standard deviation is known as the deviation of the values of data or data from the average mean of the given data. A small or less standard deviation means that the values are close and have a small gap with their average. However, larger values would indicate that the data are away from the mean value. The value of standard deviation can never be negative.
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KeyTerms: Variance, Data set, Standard Deviation, Mean, Square root, Grouped Data, Frequency Distribution.
What is Variance?
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Variance is used to calculate how far the data has scattered. When all the data values are similar, then the variance would be zero. And all other variances are considered positive. A small variance would mean that the data set values are near to each other and also the mean. The variance is said to be high when the data points are far away from one another and the mean.
What is Standard Deviation?
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Standard deviation denotes the deviation from the mean. It helps to understand the dispersion from the mean.
Standard deviation is also similar to variance, when the data points are nearer to the mean, the variation is less. And when data points are a little far from the mean, then there's a huge variance.
Standard deviation
How to obtain Standard Deviation v/s Variance?
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- Variance is obtained by calculating the mean of the data set and by subtracting the mean from each data value individually.
- Further, square the output and take the mean of the obtained squared values.
- Standard deviation is the square root of the variance output.
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Variance and Standard Deviation Formula
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The given below are formulas for variance and standard deviation. generally, it is denoted by sigma (σ).
The Standard Deviation Formula:
xi = Data set values
x-= Mean of the data
The Formula for Variance:
Variance,
Also, ‘σ’ denotes the standard deviation. 'X' denotes every single value of the population or data. ‘μ ’ denotes the mean of all the data points and ‘n’ denotes the total number of values.
How to Calculate the Standard Deviation?
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In general, the results of standard deviation are very influential. It helps in measuring dispersion. The standard deviation has three important features. The primary feature is measured by using the arithmetic mean method.
- The deviation is measured by considering the mean of the reference.
- Second, it works with positive values. And lastly, the standard deviation value is generally positive as it's a square root.
- Below is the step-by-step process of measuring standard deviation.
Steps to Calculate the Standard Deviation
- First add all the data values and divide by the count of data points, and obtain the mean value.
- Then obtain the variance by subtracting the value of data points from the mean.
- In the next step, square the calculated values and add the results.
- Later, divide the results by the count of data points.
- Finally, take the square root of the variance from the preceding step. And the obtained value is the required standard deviation.
Standard Deviation is of two different types:
- Population Standard Deviation
- Sample Standard Deviation
Formula to obtain different Standard Deviations:
| Population Standard Deviation Formula | σ=∑(X−μ)2n |
| Sample Standard Deviation Formula | s=∑(X−X¯)2n−1 |
Notations for Standard Deviation
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- σ = Standard Deviation
- xi = points Given in the Data
- x̄ = Mean of the values
- n = Total number of data points
Standard Deviation Formula for Discrete Frequency Distribution
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To find the discrete frequency distribution of the data:
x: x1, x2, x3, … xn and
f: f1, f2, f3, … fn
The formula for standard deviation is:
where N is provided as:
N = n∑i=1 fi
Standard Deviation Formula based on Grouped Data
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There is a different standard deviation formula that is obtained from the variance. This formula is denoted as:
Where N = n∑i=1 fi
Standard Deviation Formula based on Non-Grouped Data
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Non-grouped data is just a random list of values. The standard deviation is obtained by the formula:
σ=√1N∑Ni=1(Xi−μ)2
Where N = n∑i=1 fi
Things to remember
- The standard deviation is always positive.
- Standard deviation denotes the gap between points.
- Standard deviation and variance are similar but not the same.
- There are different formulas for different types of data provided to find the standard deviation.
- The variance is said to be high when the data points are far away from one another and the mean.
- Standard deviation denotes the deviation from the mean.
Sample questions
Ques. Calculate the mean, variance, and standard deviation for the following data: (5 Marks)
Ans.
| Class Interval | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 |
|---|---|---|---|---|---|---|
| Frequency | 27 | 10 | 7 | 5 | 4 | 2 |
| Class Interval | Frequency (f) | Mid Value (xi) | fxi | fxi2 |
|---|---|---|---|---|
| 0 – 10 | 27 | 5 | 135 | 675 |
| 10 – 20 | 10 | 15 | 150 | 2250 |
| 20 – 30 | 7 | 25 | 175 | 4375 |
| 30 – 40 | 5 | 35 | 175 | 6125 |
| 40 – 50 | 4 | 45 | 180 | 8100 |
| 50 – 60 | 2 | 55 | 110 | 6050 |
| ∑f = 55 | ∑fxi = 925 | ∑fxi2 = 27575 |
By following the steps, we obtain from the question as follows:
N = ∑f = 55
Mean = (∑fxi)/N = 925/55 = 16.818
Variance = 1/(N – 1) [∑fxi2 – 1/N(∑fxi)2]
= 1/(55 – 1) [27575 – (1/55) (925)2]
= (1/54) [27575 – 15556.8182]
= 222.559
Standard deviation = √variance = √222.559 = 14.918
Ques. How do we obtain the standard deviation? (4 Marks)
Ans. The steps to obtain the standard deviation are as given below:
Step 1: Find the mean for the given data set.
Step 2: Subtract the mean from each data point and find the square in each situation.
Step 3: Calculate the mean of the obtained squared deviations.
Step 4: Lastly, take the square root obtained mean to obtain the standard deviation.
Ques. What does Standard Deviation provide us? (1 mark)
Ans. The standard deviation shows us how much gap the mean has from each observation in the provided data set. In other words, it presents the basic deviation from the mean.
Ques. What are the standard deviations and variance? (2 marks)
Ans. Standard deviation represents how the spread of observations of a data set is from the mean by observing at the variance’s square root. The variance calculates how the average degree to which each observation varies from the mean of all observations of the provided data.
Ques. Provide the standard deviation example. (2 marks)
Ans. Given the data set: 2, 1, 3, 2, 4. The mean and the sum of squares of deviations of the provided observations from the mean would be 2.4 and 5.2, respectively.
Hence, the standard deviation will be √(5.2/5), which results in 1.01.
Ques. When should we use standard deviation? (1 Mark)
Ans. The standard deviation is used in estimations for a group of observations (i.e., data provided) are scattered out from the mean of the data (average or expected value).
Ques. Find the standard deviation of the given numbers : 3, 8, 6, 10, 12, 9, 11, 10, 12, and 7. (5 Marks)
Ans. Step 1: Firstly, we have to find the mean of the provided ten data values which is
X-= (3 + 8 + 6 + 10 + 12 + 9 + 11 + 10 + 12 + 7)/ 10 = 88 / 10 = 8.8
Step 2: We need to make a table as below with three different columns containing, One column which holds the values of x, the next second column which holds the deviation values, and the third which has squared deviations.
| Value (x) | X – X- | ( x – x-)2 |
|---|---|---|
| 3 | -5.8 | 33.64 |
| 8 | -0.8 | 0.64 |
| 6 | -2.8 | 7.84 |
| 10 | 1.2 | 1.44 |
| 12 | 3.2 | 10.24 |
| 9 | 0.2 | 0.04 |
| 11 | 2.2 | 4.84 |
| 10 | 1.2 | 1.44 |
| 12 | 3.2 | 10.24 |
| 7 | -1.8 | 3.24 |
| Total | 0 | 73.6 |
Step 3: Since the data is not in the form of sample data, we need to make use of the population variance formula.σ2=1NN∑i=1(xi−μ)2
The standard deviation formula is, Thus we have1(xi−μ)2
σ = √ 73.6 / 10 = √ 7.36.
Lastly, the standard deviation we obtain is 2.71.
Ques. Find the standard deviation of 4, 9, 11, 12, 17, 5, 8, 12, 14 (4 Marks)
Ans.
First find out the mean: 10.222
Now, subtract the mean individually from each of the data set points provided and then square the obtained result, which is equivalent to the (x - μ)² step.
x would refer to the values given in the question.
| X | 4 | 9 | 11 | 12 | 17 | 5 | 8 | 12 | 14 |
| (x-)2 | 38.7 | 1.49 | 0.60 | 3.16 | 45.9 | 27.3 | 4.94 | 3.16 | 14.3 |
Now, add up the obtained results (which is the 'sigma' in the formula): 139.55
Now, divide by n. as discussed earlier n is the number of values in the data, so in this example, N is 9. which gives us: 15.51
And lastly, square root it: 3.94
Ques. When we have the grouped data, such as the following: (3 Marks)
Ans.
| X | f |
|---|---|
| 4 | 9 |
| 5 | 14 |
| 6 | 22 |
| 7 | 11 |
| 8 | 17 |
the formula for standard deviation used:
By finding the variances of the values and then the square root. Obtain the final result i.e., standard deviation as 1.32.
Ques. Consider the number of gold coins 5 pirates have; 4, 2, 5, 8, 6. Find the standard deviation. (5 Marks)
Ans. Mean:
x¯=∑xn
=x1+x2+x3+x4…..+xnn
= (4 + 2 + 5 + 6 + 8) / 5
= 5
xn−x¯ for every value of the sample:
x1−x¯=4–5=−1
x2−x¯=2–5=−3
x3−x¯=5–5=0
x4−x¯=8–5=3
x5−x¯=6–5=1
∑(xn−x¯)2
=(x1−x¯)2+(x2−x¯)2+…+(x5−x¯)2
=(−1)2+(−3)2+02+32+12
= 20
Standard deviation:
S.D=∑(in−x¯)2n−1
=204
= √5
= 2.236
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