Statistics: Definitions, Types, Formulas, Applications & Importance

Statistics in Mathematics

Statistics is a branch of mathematics dealing with data and its interpretation. Statistics is the science of counting, averages, estimates, and probability. Statistics are of two types, namely, Descriptive Statistics and Inferential Statistics.

Descriptive Statistics

Descriptive Statistics is about organising and summarising the collected data to understand its meaning and purpose.

Inferential Statistics

Inferential Statistics is about predicting and drawing a conclusion from the organised data.

For example, In a set of marks of a student population, descriptive statistics is finding the average marks, the range of marks from highest to lowest, etc. while inferential statistics is about giving a conclusion like comparing this set of marks with the previous test marks to find the improvement.

The interpretation of data using statistics is mainly based on standard formulas. From mean, median, and mode to variance and probability, these statistics formulas help in analyzing and understanding different types of data.

What is Data?

Any bit of information is called data. For example, the average marks you scored in an exam or the number of goals in a football match are data points. In simple words, statistics meaning is about collecting and organizing such data to conclude.

Types of Data

Data can be classified into two types:

1. Quantitative Data – Numeric data that can be counted or measured (e.g., height, marks, temperature).

2. Qualitative Data – Non-numeric data that describes qualities or characteristics (e.g., colors, opinions, feedback).

Methods of Data Collection

The collection of data is the first step in statistics. It can be done through:

1. Surveys – Asking individuals questions to collect information.

2. Experiments – Collecting data under specific controlled conditions.

3. Observational Studies – Collecting data by simply observing without interference.

Representation of Data

The data once collected must be arranged or organized in a way so that inferences or conclusions can be made out from it.

The following are the ways for representation of data

1. Ungrouped distribution
2. Ungrouped frequency distribution
3. Grouped frequency distribution

Frequency Distribution in Statistics

The frequency of a data point represents the number of times it appears in a dataset. To make data more readable, we often use different types of frequency distributions in statistics, such as ungrouped distribution, ungrouped frequency distribution, and grouped frequency distribution.

1- Ungrouped Distribution

In an ungrouped distribution, all the values are written individually, separated by commas, without organizing them into classes.

Example:
Marks obtained (out of 100) by 30 Class XI students:

$10, 20, 36, 92, 95, 40, 50, 56, 60, 70,$
$92, 88, 80, 70, 72, 70, 36, 40, 36, 40,$
$92, 40, 50, 50, 56, 60, 70, 60, 60, 88$

This is called an ungrouped distribution because the data is simply listed without further organization.

2- Ungrouped Frequency Distribution

Instead of repeating values, we can show how many times each value occurs. For example, in the dataset above, 4 students scored 70 marks, so the frequency of 70 is 4.

This can be shown in a table called an Ungrouped Frequency Distribution:

3- Grouped Frequency Distribution

To make large data more manageable, we can arrange it into class intervals (ranges) and record the number of values in each range.

Example:

Here, the class width is $15$ (for example, $25 - 10 = 15$, $70 - 55 = 15$). The width can be chosen based on convenience.

Measures of Central Tendency

It is often convenient to have a single number that represents the whole data. Such a number is called a Measure of Central Tendency. This value usually lies near the middle of the data and helps in understanding the overall pattern. The three most common measures of central tendency are Mean, Median, and Mode.

• Mean
• Median
• Mode

Mean

The mean of given values is the sum of all observations divided by the total number of observations.

If we have $n$ values $x_1, x_2, x_3, \ldots, x_n$, then the mean $\bar{x}$ (read as “x-bar”) is: $\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}$

Example: To calculate the mean marks of 50 students, add the marks of all 50 students and divide by 50.

Mean of Ungrouped Data

If $n$ observations are $x_1, x_2, x_3, \ldots, x_n$, then: $\bar{x} = \frac{x_1+x_2+\cdots+x_n}{n} = \frac{1}{n} \sum_{i=1}^n x_i$

Mean of Ungrouped Frequency Distribution

If values are $x_1, x_2, x_3, \ldots, x_n$ with respective frequencies $f_1, f_2, f_3, \ldots, f_n$, then: $\bar{x} = \frac{f_1x_1+f_2x_2+f_3x_3+\cdots+f_nx_n}{f_1+f_2+f_3+\cdots+f_n} = \frac{\sum_{i=1}^n f_i x_i}{\sum_{i=1}^n f_i}$

Median

The median is the middle value of data when arranged in ascending or descending order. It divides the dataset into two equal halves.

Example:
Data: $65, 55, 89, 56, 35, 14, 56, 55, 87, 45, 92$

Arranged order: $14, 35, 45, 55, 55, 56, 56, 65, 87, 89, 92$

The middle value is $56$, so the median = 56.

If $n$ is even, the median is the average of the two middle values.

Median of Ungrouped Data

If the number of observations is $n$:

• If $n$ is odd: $\text{Median} = \left(\frac{n+1}{2}\right)^{th} \text{ observation}$

• If $n$ is even: $\text{Median} = \frac{\text{Value of } \left(\frac{n}{2}\right)^{th} \text{ observation } + \text{Value of } \left(\frac{n}{2}+1\right)^{th} \text{ observation}}{2}$

Example:
Data: $1, 11.5, 6, 7.2, 4, 8, 9, 10, 6.8, 8.3, 2, 2, 10, 1$

Ordered: $1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5$

Here $n = 14$. Median = average of 7th and 8th values = $(6.8 + 7.2)/2 = 7$.

Median of Ungrouped Frequency Distribution

Let total frequency = $N$.

• If $N$ is odd: Median = the observation with cumulative frequency $\geq \frac{N+1}{2}$.

• If $N$ is even: Median = average of the values corresponding to cumulative frequencies $\geq \frac{N}{2}$ and $\geq \frac{N}{2}+1$.

Mode

The mode is the most frequently occurring value in a dataset.

Example: Data: $65, 55, 89, 56, 35, 14, 56, 55, 87, 45, 92, 55$

Here, $55$ occurs most often, so the mode = 55.

Measures of Central Tendency and Dispersion

Central tendency helps us find a single value that represents the whole data, while measures of dispersion tell us how spread out the data is. Common measures of dispersion include range, mean deviation, variance, standard deviation, and coefficients of variation.

Range

The range is the simplest measure of dispersion in statistics. It represents the difference between the largest and smallest observations in a dataset. It gives a quick idea of how spread out the data is but does not consider the distribution of values in between.

$\text{Range} = x_{\max} - x_{\min}$

Mean Deviation

Mean deviation measures the average distance of all observations from a central value (mean, median, or any chosen value). It provides a simple understanding of data spread in statistics class 10, 11, or 12.

• Ungrouped Data: $\text{M.D.}(a) = \frac{1}{n} \sum_{i=1}^n |x_i - a|$

• Mean Deviation about Mean: $\text{M.D.}(\bar{x}) = \frac{1}{n} \sum_{i=1}^n |x_i - \bar{x}|$

• Frequency Data: $\text{M.D.}(\bar{x}) = \frac{\sum_{i=1}^n f_i |x_i - \bar{x}|}{\sum_{i=1}^n f_i}$

Variance

Variance is a fundamental concept in statistics class 11 and 12 that measures the average of the squares of deviations from the mean. It tells us how far each observation is from the mean and is a key step in calculating standard deviation.

• Ungrouped Data: $\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2$

• Frequency Data: $\sigma^2 = \frac{1}{N} \sum_{i=1}^n f_i (x_i - \bar{x})^2$

Standard Deviation

Standard deviation is the positive square root of variance and gives a measure of dispersion in the same units as the data. It is widely used in inferential statistics and helps compare the spread of different datasets.

• Ungrouped Data: $\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2}$

• Frequency Data: $\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^n f_i (x_i - \bar{x})^2}$

Coefficients of Dispersion

Coefficients of dispersion are unit-free measures of variability, which allow comparison of the spread between datasets of different scales. They are often used in statistics class 10 and 11 for analyzing data consistency.

Coefficient of Range

Shows relative spread using range.

Formula: $⁡\text{C.R.} = \frac{x_{\max} - x_{\min}}{x_{\max} + x_{\min}}$

Coefficient of Mean Deviation

Expresses mean deviation relative to the mean for a normalized measure of spread.

Formula: $\text{C.M.D.} = \frac{\text{M.D.}}{\bar{x}}$

Coefficient of Standard Deviation

Measures the relative standard deviation, giving insight into data variability independent of units.

Formula: $\text{C.S.D.} = \frac{\sigma}{\bar{x}}$

Coefficient of Variation (C.V.)

Expresses standard deviation as a percentage of the mean. It is widely preferred in statistics for comparing datasets.

Formula: $\text{C.V.} = \frac{\sigma}{\bar{x}} \times 100, \quad \bar{x} \neq 0$

Important Statistics Formulae List

This section lists all the key formulas used in statistics, from mean, median, and mode to variance, standard deviation, and coefficients of dispersion. These formulas help summarize, analyze, and interpret data efficiently in both Class 11 and 12.

Statistics in Mathematics: Solved Previous Year Questions

Question 1:

The mean and standard deviation of 100 observations are 40 and 5.1, respectively. By mistake, one observation is taken as 50 instead of 40. If the correct mean and the correct standard deviation are $\mu$ and $\sigma$ respectively, then $10(\mu+\sigma)$ is equal to:

Solution:

$\begin{aligned}
& \text { Actual mean }=\mu=\frac{100(40)-50+40}{100} \\
& \mu=40-\frac{1}{10}=39.9
\end{aligned}$

Incorrect variance

$\begin{aligned} & (5.1)^2=\frac{\sum \mathrm{x}_{\mathrm{i}}^2}{100}-(\overline{\mathrm{x}})^2 \\ & \sum \mathrm{x}_{\mathrm{i}}^2=100 \times\left(40^2\right)+100(5.1)^2 \\ & \sum \mathrm{x}_{\mathrm{i}}^2=16 \times 10^4+(5.1)^2 \times 100=162601 \\ & \sigma^2=\frac{\sum \mathrm{x}_{\mathrm{i}}^2-50^2+40^2}{100}-(\mu)^2 \\ & \sigma^2=1617.01-(39.9)^2=25 \\ & \sigma=5 \\ & 10(\mu+\sigma)=10(39.9+5) \\ & =10 \times 44.9=449\end{aligned}$

Hence, the correct answer is 449.

Question 2:

If the mean and the variance of $6,4, a, 8, b, 12,10$, 13 are 9 and 9.25 respectively, then $a+b+a b$ is equal to:

Solution:

Given the data set: $6, 4, a, 8, b, 12, 10, 13$

Mean $(\bar{x}) = 9$, Variance $(\sigma^2) = 9.25$

Number of terms, $n = 8$

First, write the equation for the mean:

$
\frac{6 + 4 + a + 8 + b + 12 + 10 + 13}{8} = 9
$

$
\Rightarrow \frac{53 + a + b}{8} = 9
$

$
\Rightarrow 53 + a + b = 72
$

$
\Rightarrow a + b = 19 \quad \ldots (1)
$

Next, write the equation for variance:

$
\sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2
$

Calculate $\sum x_i^2$:

$
6^2 + 4^2 + a^2 + 8^2 + b^2 + 12^2 + 10^2 + 13^2 = 36 + 16 + a^2 + 64 + b^2 + 144 + 100 + 169 = 529 + a^2 + b^2
$

Now use the variance formula:

$
9.25 = \frac{529 + a^2 + b^2}{8} - 9^2
$

$
9.25 = \frac{529 + a^2 + b^2}{8} - 81
$

Multiply both sides by 8:

$
74 = 529 + a^2 + b^2 - 648
$

$
74 = a^2 + b^2 - 119
$

$
a^2 + b^2 = 193 \quad \ldots (2)
$

From equations $(1)$ and $(2)$, recall:

$
(a + b)^2 = a^2 + 2ab + b^2
$

Substitute:

$
19^2 = 193 + 2ab
$

$
361 = 193 + 2ab
$

$
2ab = 168
$

$
ab = 84 \quad \ldots (3)
$

Now find $a + b + ab$:

$
a + b + ab = 19 + 84 = 103
$

Hence, the value of $a + b + ab$ is $103$.

Therefore, the correct answer is 103.

Question 3:

Let the mean and the standard deviation of the observation $2,3,3,4,5,7, \mathrm{a}, \mathrm{b}$ be 4 and $\sqrt{2}$ respectively. Then the mean deviation about the mode of these observations is:

Solution:

$\text { Mean is given by } \frac{2+3+3+4+5+7+a+b}{8}=4 \text {. }$

$\begin{aligned} & \frac{24+a+b}{8}=4 \\ & a+b=8\ldots(1)\end{aligned}$

The standard deviation is given by $\sqrt{\frac{\sum_{i=1}^8\left(x_i-4\right)^2}{8}}=\sqrt{2}$.
Squaring both sides:

$
\frac{(2-4)^2+(3-4)^2+(3-4)^2+(4-4)^2+(5-4)^2+(7-4)^2+(a-4)^2+(b-4)^2}{8}=2
$

$\begin{aligned} &2=\frac{4+1+1+0+1+9+(a-4)^2+(b-4)^2}{8} \\ & 16=48+a^2+b^2-8 a-8 b \\ & a^2+b^2=32 \\ & 32=2 a b \\ & a b=16 \\ & a=4, b=4 \\ & \text { mode }=4 \\ & \text { mean deviation }=\frac{2+1+1+0+1+3+0+0}{8}=1\end{aligned}$

Hence, the correct answer is 1.

Question 4:

Let the Mean and Variance of five observations $x_1=1, x_2=3, x_3=a, x_4=7$ and $x_5=b, a>b$, be 5 and 10 respectively. Then the Variance of the observations $n+x_n, n=1,2, \ldots \ldots \ldots 5$ is

Solution:

Calculate the mean:

$5=\frac{1+3+a+7+b}{5}$

$\Rightarrow a+b=14$

$\frac{1+9+a^2+49-b^2}{5}-(5)^2=10$

$a^2+b^2=116$

$\Rightarrow a=10, b=4$

New observations: $2,5,13,11,9$

Var $=\frac{4+26+169+121+81}{5}-64$

Var $=80.2-64$

Var $\approx 16$
Hence, the answer is 16.

Question 5:

The variance of the numbers $8,21,34,47, \ldots, 320$, is____________.

Solution:

Given the arithmetic sequence: $ 8, 21, 34, 47, \ldots, 320 $

First term, $ a = 8 $, common difference, $ d = 13 $

Use the $ n^{th} $ term formula:
$
a_n = a + (n-1)d
$

Put $ a_n = 320 $:
$
320 = 8 + (n-1) \times 13 \\
312 = 13(n-1) \\
n-1 = \frac{312}{13} = 24 \\
n = 25
$

Number of terms, $ n = 25 $

Mean of the AP:
$
\bar{x} = \frac{a + l}{2} = \frac{8 + 320}{2} = 164
$

Variance of AP is given by:
$
\sigma^2 = \frac{(n^2 - 1)d^2}{12}
$

Substitute values:
$
\sigma^2 = \frac{(25^2 - 1) \times 13^2}{12} = \frac{624 \times 169}{12}
$

Calculate numerator:
$
624 \times 169 = 105456
$

Divide:
$
\sigma^2 = \frac{105456}{12} = 8788
$

Hence, the correct answer is 8788.

List of Topics related to Statistics according to JEE Mains/NCERT

This section covers the important Statistics topics in Class 11, including data collection, presentation, measures of central tendency, and dispersion as per the NCERT and JEE Main syllabus.

Statistics in Different Exams

The chapter Statistics deals with the collection, analysis, and interpretation of numerical data. It is an important part of Class 12 Mathematics and is frequently tested in board and competitive examinations. Questions from this chapter assess a student’s ability to apply statistical formulas, analyse data sets, and interpret results accurately. Regular practice of NCERT examples and numerical problems helps students master this chapter and score well in examinations.

Important Books and Resources for Statistics

Here you’ll find the best books and resources that explain Statistics concepts clearly, with examples and practice problems to strengthen your basics.

NCERT Resources for Statistics

NCERT Class 11 Mathematics book provides detailed explanations, solved examples, and exercises that form the core foundation of Statistics.

• NCERT Maths Notes for Class 11th Chapter 15 Statistics

• NCERT Maths Solutions for Class 11th Chapter 15 Statistics

• NCERT Maths Exemplar Solutions for Class 11th Chapter 15 Statistics

NCERT Subjectwise resources

This section provides subject-specific NCERT materials such as notes, exemplar problems, and detailed solutions across different subjects, helping students strengthen their concepts along with related topics.

Practice Questions based on Statistics

This section brings you practice questions designed to test your understanding of formulas and concepts, helping you prepare for both school exams and competitive tests.

Conclusion

The chapter Statistics helps students understand how numerical data can be collected, organised, and analysed to draw meaningful conclusions. By studying measures of central tendency and dispersion, students develop strong analytical and interpretation skills. This chapter forms a solid foundation for higher studies in mathematics, economics, data science, and research-based fields. With regular practice of formulas, examples, and real-life data problems, students can master Statistics and confidently apply it in both examinations and everyday decision-making.