Measures of Dispersion Notes for SSC Statistics

Measures of dispersion show how much the values of a data set are spread around the central value. The mean, median and mode tell the centre of the data, but dispersion tells how scattered the data is. This chapter is important in the SSC CGL Statistics paper. These measures of dispersion notes for SSC explain every measure with its formula, properties and solved examples in simple language.

Types of Measures of Dispersion

Variance and Standard Deviation

The standard deviation was named by Karl Pearson. It is the best measure of dispersion because it uses all the values and is easy to use in further calculations.

Properties of Standard Deviation

• SD does not change if the same number is added to or subtracted from every value (change of origin).
• If every value is multiplied by a number a, the SD is multiplied by |a| (change of scale).
• The SD of a constant (all values same) is zero.
• SD is always positive and has the same unit as the data, and the variance has the square of that unit.
• For the first n natural numbers, the mean is (n + 1) / 2 and the variance is (n² − 1) / 12.

Solved Examples

Example 1. Find the range and its coefficient for 12, 7, 20, 15 and 9.
Range = 20 − 7 = 13. Coefficient = 13 / 27 = 0.48.

Example 2. Find the mean deviation about the mean for 3, 5, 7, 9 and 11.
Mean = 7. The absolute deviations are 4, 2, 0, 2, 4, with a sum of 12. MD = 12 / 5 = 2.4.

Example 3. Find the variance and SD of 2, 4, 4, 4, 5, 5, 7, 9.
Mean = 40 / 8 = 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4, 16, with a sum of 32. Variance = 32 / 8 = 4 and SD = 2.

Example 4. The mean of a data set is 50 and its SD is 10. Find the CV.
CV = (10 / 50) × 100 = 20 per cent.

Example 5. If the SD of x is 4, find the SD of y = 3x + 5.
The SD becomes 3 × 4 = 12, because adding 5 does not change it.

Comparing Two Series

• To compare the spread of two data sets with different units or means, use the coefficient of variation.
• The series with the lower CV is more consistent (more stable), and the series with the higher CV is more variable.

Relation Between the Measures

Quick Tips

• Use the short formula of variance for large values to reduce calculation.
• Remember that adding a number does not change the SD, but multiplying does.
• Use CV to compare two series, and the lower CV means more consistency.
• Check that the variance is positive and that SD is the square root of the variance.

Quick Revision Points

• Range = largest minus smallest.
• QD = (Q3 − Q1) / 2.
• Variance = Σx² / n − (mean)².
• SD is the square root of the variance.
• CV = (SD / mean) × 100.

Practice Questions With Answers

1. Find the range of 5, 12, 8, 20 and 15.
   Answer: 15
2. What is the SD if the variance is 25?
   Answer: 5
3. What is the SD of the data 7, 7, 7, 7?
   Answer: 0
4. If SD of x is 6, find the SD of 2x + 3.
   Answer: 12
5. The mean is 40 and SD is 8. Find the CV.
   Answer: 20 per cent
6. Which measure of dispersion is the best?
   Answer: Standard deviation
7. Find the QD if Q1 = 20 and Q3 = 50.
   Answer: 15
8. What is the variance of the first 5 natural numbers?
   Answer: 2
9. Which series is more consistent: CV 12 per cent or CV 18 per cent?
   Answer: The series with CV 12 per cent
10. What is the unit of variance if the data is in kg?
    Answer: kg squared

How to Practise This Chapter

Solve one problem of each measure daily, and write the formula before you start. Then attempt timed sets from our practice papers and the previous year papers page. Read the previous chapters, Mean, Median and Mode and Correlation and Regression, and the next chapter, Probability Distributions. See all topics on the Statistics notes page.

Frequently Asked Questions

What is dispersion?

Dispersion is the spread or scattering of the values of a data set around the central value.

What are the measures of dispersion?

They are the range, quartile deviation, mean deviation and standard deviation, along with the variance.

What is the formula of standard deviation?

Standard deviation is the square root of the average of the squared deviations from the mean.

What is the difference between variance and SD?

Variance is the average of the squared deviations, and SD is its square root. SD has the same unit as the data.

What is the coefficient of variation?

It is the SD divided by the mean, multiplied by 100. It is used to compare the spread of two series.

Which measure of dispersion is the best?

The standard deviation is the best, because it uses all the values and is used in further analysis.

What happens to SD if a number is added to all values?

The SD does not change when the same number is added to or subtracted from all the values.

What happens to SD if all values are multiplied by a number?

The SD is multiplied by the absolute value of that number.

What is the SD of a constant data?

The SD of data in which all the values are the same is zero.

What is quartile deviation?

Quartile deviation is half of the difference between the third and the first quartile.

What does a low coefficient of variation mean?

It means that the data is more consistent or stable.

How many questions come from dispersion in SSC Statistics?

Usually two to four questions come from measures of dispersion, but the number changes from year to year.