Practice Questions on Statistics Advanced (original) (raw)

Last Updated : 23 Jul, 2025

**Statistics is a branch of mathematics that involves the collection, organization, analysis, interpretation, and presentation of data. It helps us understand and interpret numerical information by transforming raw numbers into meaningful insights, and is used to make informed decisions in various fields. Statistics is the backbone of data-driven decision-making across industries and disciplines.

Important Formulas in Statistics

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Practice Questions on Statistics - Advanced

**Question 1: **Find the variance and standard deviation of the following dataset: 6, 8, 10, 12, 14
**Solution:

**Calculate the mean:

Mean (\bar{x}) = \frac{\sum x_i}{N} = \frac{6 + 8 + 10 + 12 + 14}{5} = 10

Compute squared deviations:

(x_i - \bar{x})^2 = (6-10)^2, (8-10)^2, (10-10)^2, (12-10)^2, (14-10)^2 = 16, 4, 0, 4, 16

Variance =\frac{\sum(x_i - \bar{x})^2}{N} = 16+4+0+4+16 / 5 ​=8

Standard Deviation = \sqrt{\text{Variance}} = \sqrt{8} = 2.83

Variance = 8, Standard Deviation = 2.83

**Question 2: **Find the variance and standard deviation for the following grouped data:

Class Interval Frequency
10–20 5
20–30 7
30–40 8
40–50 10
50–60 5

**Solution:

**Compute midpoints (x i ​): xi=15, 25, 35, 45, 55

**Compute mean(\bar{x}) = \frac{\sum f_i x_i}{\sum f_i} = \frac{5(15) + 7(25) + 8(35) + 10(45) + 5(55)}{35} = \frac{1275}{35} = 36.43

**Compute (x_i - \bar{x})^2 and f_i (x_i - \bar{x})^2:

xi fi x_i - \bar{x} (x_i - \bar{x})^2 f_i (x_i - \bar{x})^2
15 5 -21.43 459.26 2296.3
25 7 -11.43 130.67 914.69
35 8 -1.43 2.04 16.33
45 10 8.57 73.45 734.5
55 5 18.57 344.79 1723.95

\sum f_i (x_i - \bar{x})^2 = 5685.77

**Question 3: **The mean of a dataset is 50 and its standard deviation is 5. Calculate the coefficient of variation.
**Solution:

Coefficient of Variation (CV) = \frac{\text{SD}}{\text{Mean}} \times 100 = \frac{5}{50} \times 100 = 10\%

**Question 4 : **Find the combined standard deviation of two groups of students which have the following data:

**Solution:

**Combined mean: \bar{x}_c = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2} = \frac{5(10) + 10(15)}{15} = \frac{200}{15} = 13.33

**Combined variance: \sigma_c^2 = \frac{n_1(\sigma_1^2 + (\bar{x}_1 - \bar{x}_c)^2) + n_2(\sigma_2^2 + (\bar{x}_2 - \bar{x}_c)^2)}{n_1 + n_2}

\sigma_c^2 = \frac{5(2^2 + (10 - 13.33)^2) + 10(3^2 + (15 - 13.33)^2)}{15}

\sigma_c^2 = \frac{5(4 + 11.11) + 10(9 + 2.78)}{15} = \frac{109.45}{15} = 7.3

**Combined standard deviation: \sigma_c = \sqrt{\sigma_c^2} = \sqrt{7.3} = 2.7

**Question 5: **Find the quartile deviation for the dataset: 5, 7, 8, 10, 12, 15, 18
**Solution:

Arrange data in ascending order

Q_1 = \frac{\text{(n+1)} \cdot 1}{4} \text{th value} = \frac{8}{4} = 2\text{nd value} = 7
Q_3 = \frac{\text{(n+1)} \cdot 3}{4} \text{th value} = \frac{24}{4} = 6\text{th value} = 15

**Quartile Deviation:
QD= (Q3−Q1)/2 = (15−7)/2 = 4

**Question 6: **Find the mean deviation about the mean for the dataset: 3, 6, 9, 12, 15
**Solution:

**Mean: \bar{x} = \frac{\sum x_i}{N} = \frac{3 + 6 + 9 + 12 + 15}{5} = 9

**Deviations: |x_i - \bar{x}| = |3 - 9|, |6 - 9|, |9 - 9|, |12 - 9|, |15 - 9| = 6, 3, 0, 3, 6

**Mean Deviation: \text{MD} = \frac{\sum |x_i - \bar{x}|}{N} = \frac{6 + 3 + 0 + 3 + 6}{5} = 3.6

**Question 7: Find the variance for the sample: 5, 7, 9, 11, 13

**Solution:

**Mean: \bar{x} = \frac{5 + 7 + 9 + 11 + 13}{5} = 9

**Variance: = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{(5-9)^2 + (7-9)^2 + (9-9)^2 + (11-9)^2 + (13-9)^2}{5-1}= \frac{16 + 4 + 0 + 4 + 16}{4} = 10

**Question 8: Find the Z-score for x = 80 if the dataset has a mean of 50 and a standard deviation of 15.
**Solution:

**Given,
**x = 80
**mean\mu = 50
**Standard deviation \sigma = 15

Z = \frac{x - \mu}{\sigma} = \frac{80 - 50}{15} = \frac{30}{15} = 2

**Question 9 : **Find the range and interquartile range (IQR) for the following dataset: **8, 12, 16, 20, 24, 28, 32, 36
**Solution:

**Range = \text{Max} - \text{Min} = 36 - 8 = 28

**Quartiles:
Q1 = \text{Median of first half} = \text{Median of } [8, 12, 16, 20] = 14
Q3 =\text{Median of second half} = \text{Median of } [24, 28, 32, 36] = 30

**IQR = Q3 - Q1 = 30 - 14 = 16

**Question **10: Find the percentile rank of x=18 in the dataset: 10, 12, 15, 18, 20, 25
**Solution:

P = \frac{\text{Number of values below } x}{\text{Total number of values}} \times 100 = \frac{3}{6} \times 100 = 50

Unsolved Practice Questions on Statistics

**Question 1: A dataset contains the following values: 4, 7, 9, 10, 12, 14, 15, 18, 20, 25. Find the variance, standard deviation, and coefficient of variation.

**Question 2: Calculate the Pearson correlation coefficient and interpret the result. the following paired data:

**X = [1, 2, 3, 4, 5], Y = [2, 4, 6, 8, 10]

**Question 3: Find the combined standard deviation for two groups of students with the following details:

**Question 4: **Calculate the quartile deviation for the following dataset: 5, 8, 12, 15, 18, 22, 25, 28, 30, 35.

**Question 5: **Find the mean deviation about the mean for the dataset: 10, 12, 14, 16, 18, 20, 22, 24.

**Question 6: For the dataset 15, 18, 20, 24, 28, 30. calculate:

**Question 7: In a sample of size 7 with values 6, 8, 10, 12, 14, 16, 18 calculate the Z-score for x = 16.

**Question 8: For a grouped dataset with the following details, find the coefficient of variation:

Class Interval Frequency (fi)
0–5 3
5–10 5
10–15 4
15–20 6
20–25 2

**Question 9: A dataset has a mean of 505050 and a standard deviation of 101010.

**Question 10 : For a dataset 3, 6, 9, 12, 15, 18, 21:

Answer key for Unsolved Questions

**Answer 1:

**Answer 2: Correlation coefficient r = 1. Perfect positive linear relationship.

**Answer 3:

**Answer 4:

**Answer 5: Mean Deviation: 4

**Answer 6:

**Answer 7: Z-score = 0.670.670.67

**Answer 8:

**Answer 9:

**Answer 10:

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