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Standard Deviation

Standard Deviation: Explanation and Calculations of Sample and Population STD

Standard Deviation is a statistical terminology used to find or measure the amount of scatterness around an average of under-observation data. Dispersion or scatterness is the difference between the original and average values. The more maximize this dispersion or variability is, the more bigly the value of the standard deviation.

Standard Deviation may be represented by S.D and is most commonly symbolized in mathematics by the small alphabet (a Greek letter) sigma σ, for the population S.D.

The standard Deviation of a random variable or sample is also calculated by taking the square root of its variance. Finding it by taking the square root is simpler algebraically.

Both the standard deviation of the data and the standard error of the estimate are frequently discussed (as a measure of possible mistakes in the findings).

What is the Standard Deviation?

In statistical investigations, the measurement of variability known as the standard deviation (SD) is often and extensively employed. It reveals the degree of deviation from the data's average (mean). A low SD suggests that the data points tend to be near the mean, whereas a high SD suggests that the data are dispersed throughout a wide range of values, depending on two possible situations.

Types of Standard Deviation:

As we have discussed that the process of taking Standard deviation, it has two types.

·       Population standard deviation

·       Sample standard deviation

Population Standard Deviation:

How much variance there is between specific data points in a population is shown by the population standard deviation. We may sum it up by saying that it is a means to measure how dispersed the data is from the mean. When the numbers you have are for the entire population, it is relevant.

σ = √ {(xi - µ) 2 / N}

Sample Standard Deviation:

A sample standard deviation is a statistical measure that is computed from only a few values in a reference population. It pointed towards the standard deviation of the sample rather than that of a population.

S = √ {(xi - µ) 2 / (N – 1)}

Examples of Sample and Population STD

Example 1: (For population standard deviation)

Four colleagues were comparing their scores on a recent test.

Calculate the standard deviation of their scores: 5,7,6,2

Solution:

Step 1: Find the mean.

Mean (µ) = 5+7+6+2/4

= 20/4

= 5

The mean is 5 points.

Step 2: Each score will subtract the mean of the data.

Score (xi)

Deviation (xi  µ )

5

5 - 5 = 0

7

7 - 5 = 2

6

6 - 5 = 1

2

2 - 5= - 3

Step 3: Square each deviation.

Score (xi)

Deviation (xi  µ )

(xi  µ )2

5

5 - 5 = 0

0

7

7 - 5 = 2

4

6

6 - 5 = 1

1

2

2 - 5= - 3

9

Step 4: Add the squared deviations.

0+4+1+9=14

Step 5:  Division by number of scores.

=√14/4

=3.5

Step 6: To Take the square root of the result from Step 5.

=√3.5

=1.87

The standard deviation is approximately 1.87

Example2: (For sample standard deviation)

 Here is the data of the recent score performance of tail-ender batsman:

9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 14

Calculate the sample standard deviation of a score of a tail-ender batsman in 20 matches he played in his overall career.

Solution:

Step 1: To Calculate the mean of the data. Sum all the numbers and divide it by the total number of terms in the data.

Mean (µ) = (9 + 2 + 5 + 4 + 12 + 7 + 8 + 11 + 9 + 3 + 7 + 4 + 12 + 5 + 4 + 10 + 9 + 6 + 9 + 14)/20

   = 150/20

               = 7.5

Step 2: Subtract every number of data from the mean and squaring.

(9 - 7)2 = (2)2 = 4

(2 - 7)2 = (-5)2 = 25

(5 - 7)2 = (-2)2 = 4

(4 - 7)2 = (-3)2 = 9

(12 - 7)2 = (5)2 = 25

(7 - 7)2 = (0)2 = 0

(8 - 7)2 = (1)2 = 1

(11 - 7)2 = (4)2 = 16

(9 - 7)2 = (2)2 = 4

(3 - 7)2 = (-4)2 = 16

(7 - 7)2 = (0)2 = 0

(4 - 7)2 = (-3)2 = 9

(12 - 7)2 = (5)2 = 25

(5 - 7)2 = (-2)2 = 4

(4 - 7)2 = (-3)2 = 9

(10 - 7)2 = (3)2 = 9

(9 - 7)2 = (2)2 = 4

(6 - 7)2 = (-1)2 = 1

(9 - 7)2 = (2)2 = 4

(14 - 7)2 = (7)2= 49

Step 3: To Calculate the mean of the squared differences.

             = (4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+49)/19

= 218/19

= 11.47

This value is the sample variance. The sample variance is 11.47

A standard deviation calculator is an alternate way to calculate standard deviation of the sample and population data values.

Applications of Standard Deviation:

Standard deviation recognizes the variation of a data set in different applications, including academia, business, finance, forecasting, manufacturing, medicine, polling, population traits, etc. It also enables us how to use different tools in finding the coefficient of variation, hypothesis testing, and confidence intervals.

1.     Finance and Economics: To gauge risk and volatility, the standard deviation is widely utilized in finance and economics. As an illustration, the standard deviation is frequently used in finance to assess the volatility of stock prices or returns. Greater price volatility is indicated by larger standard deviation readings, which might suggest increased risk.

2.      Experimental Research and Data Analysis: In experimental research and data analysis, the standard deviation is a key factor. The variability or dispersion of data points around the mean is measured using this technique. Standard deviations are frequently calculated by researchers to assess the validity of their findings.

3.      Descriptive Statistics: In descriptive statistics, the standard deviation is a key indicator of dispersion. It aids in distilling the distribution of data points within a dataset. The standard deviation enables determining whether a dataset has a larger or fewer level of variance when comparing several datasets or subsets of data.

Summary

The standard deviation is the variation in the measurement of the differences of each value taken from the mean value of data after observing its types formulas, and examples as well. Moreover, if the differences themselves were summed up, the positive will firmly balance the negative and so their sum would be zero.

Courtesy : CBSE