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.

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• Statistics In Math
• Statistics Formulas

Important Statistics formulas

• Class Interval = Upper Limit - Lower Limit

Mean (\bar x) For Ungrouped Data

• Mean ( \bar x) = \frac{x_1 + x_2 + x_3 + .....x_n}{n}

Mean (\bar x) For Grouped Data

• Direct Method : ( \bar x) = \frac{\sum f_i \ x_i}{\sum f_i}
  Where, f_i = frequency of each observation/class,
  x_i = value of observations or midpoint for each class interval.

• Assumed Mean Method : ( \bar x) = A + \frac{\sum f_i \ d_i}{\sum f_i}
  Where, f_i = frequency of each observation/class,
  d_i = deviation of each observations or midpoint of each class interval. ( d_i = x_i - A )

• Step- Deviation Method : ( \bar x) = A + \frac{\sum f_i \ u_i}{\sum f_i}. \ h
  Where, f_i = frequency of each observation/class,
  u_i = Step deviation of each observations or midpoint of each class interval. ( d_i = x_i - A )

Median For Ungrouped Data

• For odd number of Observations : Median = [\frac{(n + 1)}{2}]^{th} \ term
  Where, n= number of observations.

• For even number of observations : Median = \frac{[(\frac{n}{2})^{th}\ term + \{ \frac{n}{2} + 1 \}^{th} \ term]}{2}
  Where, n= number of observations.

Median For Grouped Data

• Median = l + [\frac{(\frac{n}{2}- cf)}{f}] \times h
  Where, l = Lower Limit of Median Class
  n = Number of Observations
  f = Frequency of Median Class
  h = Class Size
  cf = Cumulative Frequency of Class Preceding Median Class

Mode for Ungrouped Data

• Mode = Observation with highest frequency / number of occurrence.

Mode for Grouped Data

• Mode = l + \{\frac{f_1 - f_2}{2f_1 - f_0-f_2} \} \times h
  Where, l = lower limit of the modal class.
  h = size of the class interval,
  f₁ = frequency of the modal class,
  f₀ = frequency of the class preceding the modal class, and
  f₂ = frequency of the class succeeding the modal class.

Relation between Mean Median and Mode

• Mode = 3 Median – 2 Mean

Statistics Practice Questions - Solved

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

Unsolved Practice Question on Statistics

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 for Unsolved Questions:

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