Introduction to Statistics (original) (raw)

Last Updated : 23 Jan, 2026

Statistics is a branch of mathematics concerned with collecting, organizing, analyzing, and interpreting numerical data. It is recognized as a distinct scientific discipline due to its broad applications across numerous fields, including science, economics, healthcare, and social sciences.

Here are some examples of statistical concepts in action:

**Statistics Terminologies

Some of the most common terms you might come across in statistics are:

**Types of Statistics

Statistics is the study of data; when we talk about properties of data, it comes under descriptive statistics. Meanwhile, the study of concluding data comes under inferential statistics.

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Types of statistics

Descriptive Statistics

Descriptive statistics uses data that describes the population either through numerical calculations, graphs, or tables. It provides a graphical summary of data.

It is simply used for summarizing objects, etc. There are two categories in this, as follows.

**Measure of Central Tendency

Measureof central tendency is also known as a summary statistic that is used to represent the center point or a particular value of a data set or sample set. In statistics, three common measures of central tendency are:

**1. Mean

Mean is the measure of the average of all valuesin a sample set. The size of the data set is calculated using the following formula:

**Mean = ∑xn∑nx​

**2. Median

Median is the middle value in a data set when the numbers are arranged in ascending or descending order.

The formula used to calculate the median of the data set is:

If n is Even, Median = [(n/2)th term + {(n/2 )+ 1}th term]/2

If n is Odd, Median = \frac{(n+1)}{2}

**3. Mode

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

**Measure of Variability

The measure of Variability is also known as the measure of dispersion and is used to describe variability in a sample or population. In statistics, there are three common measures of variability, as shown below:

**1. Range of Data

It is a given measure of how to spread apart values in a sample set or data set.

Range = Maximum value - Minimum value

**2. Variance

In probability theory and statistics, variance measures a data set's spread or dispersion. It is calculated by averaging the squared deviations from the mean. Variance is usually represented by the symbol σ2.

Variance measures variability. The more spread out the data, the greater the variance compared to the average.

**3. Standard Deviation

Standard Deviation is a measure of how widely distributed a set of values is from the mean. It compares every data point to the average of all the data points.

Standard Deviation Formula

s = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2}

Where:

**4. Interquartile Range (IQR)

The Interquartile Range is a measure of statistical dispersion, or how spread out the data points are in a data set. It is the range between the first quartile (Q1) and the third quartile (Q3) of a data set, which represents the middle 50% of the data.

Formula for IQR:

IQR = Q3 − Q1

Where:

**Inferential Statistics

Inferential Statistics makes inferences and predictions about the population based on a sample of data taken from the population. It generalizes a large dataset and applies probabilities to conclude.

It is simply used for explaining the meaning of descriptive statistics. It is simply used to analyze, interpret results, and draw conclusions. Inferential Statistics is mainly related to and associated with hypothesis testing, whose main target is to reject the null hypothesis.

**Types of Inferential Statistics

Hypothesis Testing

Hypothesis testing is a type of inferential procedure that takes the help of sample data to evaluate and assess the credibility of a hypothesis about a population.

Inferential statistics are generally used to determine how strong a relationship is within the sample. However, it is very difficult to obtain a population list and draw a random sample. Inferential statistics can be done with the help of various steps, as given below:

Data in Statistics

Data is the collection of numbers, words, or anything that can be arranged to form meaningful information. There are various types of data in the statistics that are added below.

Types of Data

Types of Quantitative Data

We have two types of quantitative data that include,

  1. **Discrete Data: The data that has a fixed value is called discrete data and can easily be counted.
  2. **Continuous Data: The data that has no fixed value and has a range of values is called continuous data. It can be measured.

Representation of Data

We can easily represent the data using various graphs, charts, or tables. The various types of representing data sets are:

Models of Statistics

Various models of Statistics are used to measure different forms of data. Some of the models of statistics are added below:

Solved Problems - Statistics

**Example 1: Find the mean of the data set.

xi fi
2 3
3 4
5 4
8 5

**Solution:

xi fi fixi
2 3 6
3 4 12
5 4 20
8 5 40

Mean = (Σf ixi)/Σfi
Σfixi = (6 + 12 + 20 + 40) = 78, and
Σfi = 16
⇒ Mean = 78/16 = 4.875

**Example 2: Find the median of the data set.

Cars Mileage Cylinder
Swift 21.3 3
Verna 20.8 2
Santra 19 5
i-20 15 4

**Solution:

**Data in order: 15, 19, 20.8, 21.3

**⇒ Median = (20.8 + 19) /2 = 39.8/2
**⇒ Median = 19.9

Example 3: Find the Standard Deviation of 4, 7, 10, 13, and 16.

**Solution:

Given,

Σxi = (4 + 7 + 10 + 13 + 16) = 50
⇒ Mean(μ) = Σxi/N = 50/5 = 10

Standard Deviation = √(σ) = √{∑i = 1n (xi - μ)}/N
⇒ SD = √{1/5[(4 - 10)2 + (7 - 10)2 + (10 - 10)2 + (13 - 10)2 + (16 - 10)2]}
⇒ SD = √{1/5[36 + 9 + 0 + 9 + 36] = √{1/5[90]} = √18