Statistics is an important part of Mathematics that covers Mean, Median, Mode, and Quartiles for different types of data. It also includes the calculation of Mean, Median, and Quartiles for grouped data, helping students understand and analyze numerical data using statistical measures.

For Officer Cadet Entrance Exam preparation, statistics questions help develop data analysis, calculation, and mathematical reasoning skills. This resource covers essential statistics formulas, solving methods, useful tricks, and practice questions related to Mean, Median, Mode, and Quartiles for both ungrouped and grouped data. Explore the complete Officer Cadet Exam resources and Maths for Officer Cadet for additional practice and study materials.

Table Of Contents

What is Statistics?
Basic Terms in Statistics
Statistics for Officer Cadet: Syllabus
1. Measures of Central Tendency for Ungrouped Data
2. Calculation of Mean, Median and Quartile for Grouped Data
Effective Tricks to Solve Statistics Questions
Important Tips to Solve Statistics Questions
Practice Questions
Conclusion
Related Resources

What is Statistics?

Statistics is a branch of mathematics that deals with collecting, organizing, presenting, analyzing, and interpreting data. It helps us find meaningful patterns and make informed decisions from raw information. A key part of statistics is Measures of Central Tendency, which provide a single value that represents the center or middle of a data set.

Basic Terms in Statistics

Before solving statistics questions, it is important to understand these basic terms:

Term	Meaning	Example
Data	A collection of facts, numbers, or information	The marks of students in a class
Arrayed Data	Data arranged in ascending or descending order	12, 15, 18, 21, 25
Frequency (f)	The number of times a particular value appears	If 5 students scored 80 marks, the frequency of 80 is 5
Cumulative Frequency (cf)	The running total of frequencies	The total number of students up to a certain mark range
Class Interval	A range of values in grouped data	10-20, 20-30, etc.
Class Mark	The midpoint of a class interval	For 10-20, the class mark is 15

Statistics for Officer Cadet: Syllabus

This Mathematics for Officer Cadet Entrance Exam Preparation resource covers the following main topics:
1. Mean, Median, Mode and Quartile of different data
2. Calculation of Mean, Median and Quartile of grouped data

1. Measures of Central Tendency for Ungrouped Data

Ungrouped data is raw data that has not been organized into a frequency distribution.

A. Mean (Arithmetic Mean)

The Mean is the most common measure of central tendency. It is the sum of all values divided by the total number of values.

Formula:
For ungrouped data, $\overset{ˉ}{x} = \frac{\sum x}{n}$

Where:
∑x = Sum of all observations
n = Total number of observations
Example 1:
Find the mean of the following data: 10, 15, 20, 25, 30.

Solution:
$\overset{ˉ}{x} = \frac{10 + 15 + 20 + 25 + 30}{5} = \frac{100}{5} = 20$

B. Median

The Median is the middle value of an arrayed data set. It divides the data into two equal halves.

Formula:
If n is odd: Median = Value of $(\frac{n + 1}{2})^{t h}$ term
If n is even: Median = $\frac{Value of (\frac{n}{2})^{t h} term + Value of (\frac{n}{2} + 1)^{t h} term}{2}$

Example 2 (Odd n):
Find the median of: 25, 10, 15, 35, 30.

Solution:
Arrayed data: 10, 15, 25, 30, 35.
Since n = 5 (odd), Median = Value of $(\frac{5 + 1}{2})^{t h}$ term = $3^{r d}$ term = 25.

Example 3 (Even n):
Find the median of: 26, 8, 6, 12, 15, 32.

Solution:
Arrayed data: 6, 8, 12, 15, 26, 32.
Since n = 6 (even), Median = $\frac{12 + 15}{2} = \frac{27}{2} =$ 13.5.

C. Mode

The Mode is the value that appears most frequently in a data set.

Example 4:
Find the mode of: 2, 4, 4, 6, 7, 4, 8.

Solution:
The value 4 appears 3 times, which is more than any other value. So, Mode = 4.

D. Quartiles

Quartiles divide an arrayed data set into four equal parts. There are three quartiles: $Q_{1}$ 1 (Lower Quartile), $Q_{2}$ (Median), and $Q_{3}$ (Upper Quartile).

Formula (for ungrouped data):
$Q_{1}$ Q1 = Value of $(\frac{n + 1}{4})^{t h}$ term
$Q_{2}$ Q2 = Median (Value of $(\frac{2 (n + 1)}{4})^{t h}$ term)
$Q_{3}$ Q3 = Value of $(\frac{3 (n + 1)}{4})^{t h}$ term

Example 5:
Find the quartiles for: 15, 10, 20, 25, 18, 11, 9, 12

Solution:
Arrayed data: 9, 10, 11, 12, 15, 18, 20, 25. (n = 8)
$Q_{1}$ Q1 = Value of $(\frac{8 + 1}{4})^{t h}$ term = ${2.25}^{t h}$ term.
$Q_{1} = 2^{n d} term + 0.25 \times (3^{r d} term - 2^{n d} term)$
$Q_{1} = 10 + 0.25 \times (11 - 10) = 10.25$
$Q_{2}$ Q2 (Median) = $\frac{12 + 15}{2} = 13.5$
$Q_{3}$ Q3 = Value of $(\frac{3 (8 + 1)}{4})^{t h}$ term = ${6.75}^{t h}$ term.
$Q_{3} = 6^{t h} term + 0.75 \times (7^{t h} term - 6^{t h} term)$
$Q_{3} = 18 + 0.75 \times (20 - 18) = 19.5$

2. Calculation of Mean, Median and Quartile for Grouped Data

Grouped data is arranged in a frequency distribution table.

A. Mean for Grouped Data

Formula:
$\overset{ˉ}{x} = \frac{\sum f x}{\sum f} = \frac{\sum f x}{N}$
Where:
$f$ = Frequency of each class
$x$ = Midpoint (Class Mark) of each class
$N = \sum f$ = Total frequency

Example 6:
Calculate the mean for the following data.

Class Interval	Frequency (f)
0-10	5
10-20	8
20-30	12
30-40	5

Solution:

Class	f	Midpoint (x)	fx
0-10	5	5	25
10-20	8	15	120
20-30	12	25	300
30-40	5	35	175
Total	N = 30		Σfx = 620

Mean $\overset{ˉ}{x} = \frac{620}{30} =$ 20.67.

B. Median for Grouped Data

Formula:
Median = $L + \frac{h}{f} (\frac{N}{2} - c f)$
Where:
$L$ = Lower boundary of the median class
$h$ = Class width (size of the interval)
$f$ = Frequency of the median class
$N = \sum f$ = Total frequency
$c f$ = Cumulative frequency of the class preceding the median class

Steps:
1. Find $N / 2$ .
2. Locate the median class (the class where the cumulative frequency just exceeds $N / 2$ ).
3. Apply the formula.

Example 7:
Find the median for the data in Example 6.

Solution:

Class	f	cf
0-10	5	5
10-20	8	13
20-30	12	25
30-40	5	30

$N / 2 = 30 / 2 = 15$ .
The cumulative frequency just greater than 15 is 25, so the median class is 20-30.
$L = 20$ , $h = 10$ , $f = 12$ , $c f = 13$ .
Median = $20 + \frac{10}{12} (15 - 13) = 20 + \frac{10}{12} (2) = 20 + 1.67 =$ = 21.67.

C. Quartiles for Grouped Data

Formula:
$Q_{1} = L + \frac{h}{f} (\frac{N}{4} - c f)$
$Q_{3} = L + \frac{h}{f} (\frac{3 N}{4} - c f)$

Where the terms have the same meaning as in the median formula, but applied to the respective quartile class (where cf exceeds $N / 4$ for $Q_{1}$ or $3 N / 4$ for $Q_{3}$ ).

Example 8:
Calculate $Q_{1}$ Q1 and $Q_{3}$ Q3 for the data in Example 6.

Solution:
For Q1:
$N / 4 = 30 / 4 = 7.5$
The cumulative frequency just greater than 7.5 is 13, so the $Q_{1}$ class is 10-20.
$L = 10$ , $h = 10$ , $f = 8$ , $c f = 5$ .
$Q_{1} = 10 + \frac{10}{8} (7.5 - 5) = 10 + \frac{10}{8} (2.5) = 10 + 3.125 =$ 13.125.

For Q3:
$3 N / 4 = 3 \times 30 / 4 = 22.5$ .
The cumulative frequency just greater than 22.5 is 25, so the $Q_{3}$ class is 20-30.
$L = 20$ , $h = 10$ , $f = 12$ , $c f = 13$ .
$Q_{3} = 20 + \frac{10}{12} (22.5 - 13) = 20 + \frac{10}{12} (9.5) = 20 + 7.917 =$ 27.917.

Effective Tricks to Solve Statistics Questions

Trick 1: Master the Formulas for Grouped Data
The formulas for Median and Quartiles in grouped data are very similar. Remember the core structure:
$L + \frac{h}{f} (Fraction of N - c f)$
The only thing that changes is the fraction of N: $N / 2$ for Median, $N / 4$ for $Q_{1}$ and $3 N / 4$ for $Q_{3}$ .

Trick 2: Identify the Correct Class First
Before applying the formula, always find the class that contains your target value (Median, $Q_{1}$ , etc.). This avoids using the wrong frequency ( $f$ ) or cumulative frequency ( $c f$ ).

Trick 3: Use the Empirical Relationship for Mode
When you need to find the Mode in a moderately skewed distribution and are given the Mean and Median, use the empirical formula:
Mode = 3(Median) – 2(Mean)

Trick 4: Verify the Median is Between the Correct Quartiles
A quick check: $Q_{1} < M e d i a n < Q_{3}$ . If your calculated values don’t satisfy this, there is likely a calculation error.

Trick 5: Check that Cumulative Frequency is Always Increasing
The cumulative frequency column should always be non-decreasing. This helps you spot errors in your frequency table.

Trick 6: For Ungrouped Data, Always Sort First
The most common mistake is forgetting to arrange the data in ascending order before finding the Median or Quartiles.

Important Tips to Solve Statistics Questions

👉 Read the question carefully — Determine if it’s asking for Mean, Median, Mode, or a Quartile.
👉 Write down the frequency table — For grouped data, always draw a table with columns for class, frequency, and cumulative frequency.
👉 Keep formulas handy — Know the formulas for both ungrouped and grouped data by heart.
👉 Practise with different data sets — The more you practice, the faster you’ll become at identifying the correct class and applying the formula.
👉 Simplify your final answer — Round off appropriately, as per the requirement of the question.
👉 Time yourself — Statistics questions can be time-consuming, especially with large frequency tables, so build speed through repetition.

Practice Questions

Conclusion

Statistics is a scoring and logical topic in the mathematics syllabus for Officer Cadet Entrance Exam preparation. Most statistics questions can be solved efficiently once you can correctly identify whether the data is grouped or ungrouped and apply the matching formula for Mean, Median, or Quartiles. Focus on the key formulas, practice constructing frequency tables, and use the empirical relationship for Mode. With regular practice, statistics questions can become a quick-scoring area in the Mathematics section.

This resource is part of the Officer Cadet Entrance Exam Preparation series, focusing on Maths for Officer Cadet Entrance Exam. For structured lessons, comprehensive practice question banks, and mock tests, explore the Statistics for Officer Cadet section of the Maths for Officer Cadet Entrance Exam course on the Pratima Academy LMS.

Statistics for Officer Cadet: Maths Formulas, Tricks, and Practice Questions