Statistics Practice Questions - Basic

Statistics is the branch of mathematics that deals with collecting, organizing, analyzing, interpreting, and presenting data.

• It helps us make sense of complex data and draw meaningful conclusions for decision-making or predictions.

Question 1: If the marks scored by the students in a class test out of 50 are,

a) What is the class interval of 20 - 30 ?
b) How many students scored in the range of 20 - 40?

Solution:

a) Class interval = Upper limit - Lower limit
Class interval of 20-30 = 30 - 20 = 10

b) The number of students scored in range of 20 - 40 = frequency of class 20 - 30 + frequency of class 30 - 40
The number of students scored in range of 20 - 40 = 9 + 6 = 15

Question 2: Calculate the average of the following numbers: 6, 8, 2, 3, 12, 14.
Solution:

Mean = Sum of all given observations/ Total number of observations

Total number of observations = 6
Mean = 6 + 8 + 2 + 3 + 12 + 14 / 6 = 45/6 = 7.5

Thus, the Mean of 6, 8, 2, 3, 12, 14 is 7.5

Question 3: Find the mean of the first 10 even numbers.
Solution:

Sum of first n even numbers = n × (n + 1)

Sum of first 10 even numbers = 10 × ( 10 + 1) = 10 × 11 = 110
Mean of first 10 even numbers = 110 / 10 = 11

Thus, the mean of first 10 even numbers is 11.

Questions 4: The following are the monthly salaries (in $1000s) of 15 employees in a company:
30, 45, 25, 35, 40, 50, 60, 55, 45, 50, 25, 35, 40, 45, 50.
Find the median salary.

Solution:

Sort the data: [25, 25, 30, 35, 35, 40, 40, 45, 45, 45, 50, 50, 50, 55, 60].

Total number of elements (N) = 15 (odd).
Median = \frac{N}{2}^{th}\ term = 45.

Question 5: The following table represents the marks obtained by students in a test. Find the median of the marks.

Solution:

Calculate the cumulative frequency:

Class Interval	Frequency	Cumulative Frequency
0–20	5	5
20–40	8	13
40–60	12	25
60–80	15	40
80–100	10	50

Total frequency (N) = 50.

Median class = class where \frac{N}{2}^{th}observation \frac{50}{2} = 25 exists.
Median class = 40 – 60.

Median = L + \left( \frac{\frac{N}{2} - F}{f} \right) \cdot h

Where:
L = 40 (lower boundary of the median class),
N=50 ( total number of observations )
F = 13 ( cumulative frequency before the median class ),
f = 12 (frequency of the median class),
h=20 (class width).

Substituting the values:
Median = 40 + \left( \frac{25 - 13}{12} \right) \cdot 20
Median = 40 + \left( \frac{12}{12} \right) \cdot 20 = 40 + 20 = 60

Question 6: The following table shows the marks scored by students in a test. Calculate the mean using the assumed mean method.

Solution:

Choose A=35 (assumed mean).
Calculate d_i = x_i − A , where x_i​ is the class midpoint.

Class Interval	Frequency (fi​)	Midpoint (xi​)	di=xi − A	fi⋅di
10–20	5	15	−20	−100
20–30	8	25	−10	−80
30–40	12	35	0	0
40–50	15	45	10	150
50–60	10	55	20	200

• ∑f_i​ = 50
• ∑f_id_i = 170

For assumed mean method :

Mean ( \bar x) = A + \frac{\sum f_i \ d_i}{\sum f_i}

Mean = 35 + \frac{170}{50} = 35 + 3.4 = 38.4

Question 7: The following table shows the daily wages of workers in a factory. Use the assumed mean method to calculate the mean daily wage.

Solution:

Assume A = 75 (from the midpoint of the central class, 70–80).

Find the midpoints (xi):

Class Interval	Frequency (fi)	Midpoint (xi​)	di=xi−A	fi⋅di
50–60	6	55	−20-20−20	−120-120−120
60–70	10	65	−10-10−10	−100-100−100
70–80	20	75	000	000
80–90	12	85	101010	120120120
90–100	8	95	202020	160160160

Apply the formula for the mean:

Mean = A + \frac{\sum f_i d_i}{\sum f_i}

Substitute the values:

• ∑f_i = 56
• ∑f_id_i=60

Mean = 75 + \frac{60}{56}​ = 75 + 1.07 = 76.07

Question 8: Find the mode of the following dataset representing the number of hours studied by students in a week:
6, 8, 9, 6, 7, 8, 6, 9, 7, 8, 8, 6, 7, 8, 9

Solution:

Organize the data and count the frequency of each value:

Hours Studied	Frequency
6	4
7	3
8	5
9	3

Identify the mode:
The mode is the value with the highest frequency.
Mode = 8 (Frequency: 5)

Question 9: Find the mode for the following frequency distribution of students' marks:

Solution:

Identify the modal class: The class with the highest frequency is 40–5040–5040–50, so it is the modal class.

Mode = L + \left( \frac{f_m - f_1}{2f_m - f_1 - f_2} \right) \cdot h

Where:

• L = 40 (lower boundary of the modal class),
• f_m = 18 (frequency of the modal class),
• f₁ = 12 (frequency of the class before modal class),
• f₂ = 10 (frequency of the class after modal class),
• h=10 (class width).

Mode = 40 + \left( \frac{18 - 12}{2(18) - 12 - 10} \right) \cdot 10

Mode = 40 + \left( \frac{6}{36 - 22} \right) \cdot 10

Mode = 40 + \left( \frac{6}{14} \right) \cdot 10

Mode = 40 + 4.29 = 44.29

Question 10: A dataset is positively skewed. If the mode is 40 and the median is 45, find the mean using the relationship between mean, median, and mode.

Solution:

Using the formula: Mode = 3(Median) − 2(Mean)

Mode = 3(45)−2(40)= 135 − 80 = 55

Practice Questions

Question 1: The following table represents the heights (in cm) of a group of students:

Find the class width of the intervals.

Question 2: Given the following raw data:
35, 40, 42, 45, 50, 52, 55, 60, 65, 68, 70, 72
Find the frequency of the values ranging from 40 to 60.

Question 3: The following data represents the marks scored by students in a test:

Calculate the mean using the direct method.

Question 4: The wages of workers in a factory are shown below. Use the assumed mean method to calculate the mean wage:

Question 5: The following data represents the monthly expenses (in $) of 12 families: 450, 500, 550, 600, 620, 700, 720, 750, 800, 850, 900, 950. Find the median of the dataset.

Question 6: The table below shows the marks obtained by students in a class:

Find the mode using the formula for grouped data.

Question 7: The following table represents the number of hours students spent studying:

Find the median using the formula for grouped data.

Question 8: The following dataset represents the number of books borrowed by students in a week: 2, 3, 5, 3, 4, 3, 2, 5, 3, 2, 4, 5, 3, 3, 4. Find the mode of the dataset.

Question 9: The mean and median of a dataset are given as 25 and 22, respectively. Use the relationship between mean, median, and mode to calculate the mode.

Question 10: Given the frequency distribution below, answer the following questions:

• a) What is the total frequency?
• b) Identify the modal class.
• c) Calculate the median class.

Answer key

1. 10
2. 7
3. 35.86
4. 450
5. 710
6. 46.25
7. 5.3
8. 3
9. 16
10. a) 60
    b) 20-30
    c) 30-40