categorized into two main parts

Gives an idea about the data we already have and gives a summary of all the features we can get from a given data. it consists,

Examples:

Get insights of a large population using a sample of that population, in this we examine a representative subset(sample) of the population

Probability measures how likely an event is to occur it's a way of quantifying uncertainty.

Example:

Two types

A true zero means that the zero point on the scale represents the complete absence of the quantity being measured.

0°C does not mean “no temperature,” just a point on the scale. So it has no true zeros.

used in descriptive statistics, for

Tells us where the center or typical value of the data set lies

we use following to as main measurements in central tendency :

Example:

This tells us that the data is centered around 6.

Notation:

Given the population values:

Given a sample(values unknown):

Given a sample(values known):

Simple arithmetic gives equal importance to all the observations in a data set.

But when relative importance of the items in the distribution is not the same, we need to use this

Given are samples and are the weights corresponding to the observations then,

Exercise:

A student’s final marks in Mathematics, English, Statistics and Computer are 92, 76, 95, and 80 respectively. If the respective credits received for these courses are 2,
1, 3 and 4, determine an appropriate average mark.

Simple arithmetic means of two or more related groups can be combined into a composite mean

Given are set of simple arithmetic means of related groups and are the respective sample size then,

Exercise:

There are three branches of a company, employing 100, 20, and 80 persons respectively. If the arithmetic means of the monthly salaries paid by the three
companies are Rs. 50000, 60000, and Rs. 80000 respectively, find the arithmetic mean
of the salaries of all the employees of the three companies.

Primarily used to determine overall growth rates or averages of percentages over time
This can only be used in positive numbers

Used for a set of positive observations :

(This notation means you multiply all the numbers together and then take the root, where _is the count of numbers in the set)

Exercise:

The economic growth rate of a country for the last three years is 4%, 2%, and 8%. What is the overall growth rate?

The overall economic growth rate over the three years is approximately 4.6%

This version is used when dealing with growth rates that may include negative numbers (as long as they are greater than -1)

If the values are , the alternative geometric mean is:

Exercise:

The economic growth rate of a country for the last three years is 4%, -2%,
and 8%. What is the overall growth rate?
Three yearly growth rates: 4%, -2%, 8%

The average yearly growth rate is 3.21%.

This is especially useful when dealing with rates or units, for example, calculating average speed.

For a set of observations , the harmonic mean is:

Where:

Exercise:

The speeds of three deliveries (in km per hour) of a fast bowler are 60, 90, and 100. What is the average speed?

Final Answer

For any set of positive numbers with at least two different values, the following inequality holds:

Where:

This shows that the harmonic mean is always the smallest of the three averages.

Exercise:

Calculate the simple arithmetic mean, geometric mean, and harmonic mean
of 2, 4, 4, and 8.

This confirms that the harmonic mean is the smallest, followed by the geometric mean, and the arithmetic mean is the largest.

The mode is the value of a variable that occurs most frequently in a distribution.

Types of Mode:

Exercise:

12 students in a class have the following shoe sizes. What is the mode?

The median is the middle value of a data set after arranging the observations in ascending order.

Let be the total number of observations.

The median is the value at the

position.

The median lies between the two positions:

The median is the average of the values at these two positions.

Examples:

Example 1: Odd Number of Observations

The 3rd value is:

Example 2: Even Number of Observations

Quartiles are values that divide an ordered data set into four equal parts.
Each part contains 25% of the total number of observations.

Let ( ) be the total number of observations.
Location of the quartile

Exercise:

12 students in a class have the following weights (kg):

Find and .

This lies between the 3rd and 4th items.

3rd item ( = 56 )
4th item ( = 59 )

This lies between the 6th and 7th items.

6th item ( = 67 )
7th item ( = 68 )

This lies between the 9th and 10th items.

9th item ( = 79 )
10th item ( = 81 )

When dealing with large masses of raw data, it is often convenient to classify the data into groups or classes.

Example Table – Weights of 600 University Students

If measurements are recorded to the nearest unit:

Summary Table for Example Data

For grouped data, we calculate Mean, Mode, Median, Quartiles, Deciles, and Percentiles using class midpoints, class frequencies, and cumulative frequencies.

Let ( ) be the midpoints of ( ) class intervals,
and ( ) be their frequencies.

The mean is:

Example Table:

The mode is the value that occurs most frequently. For grouped data, the modal class has the highest frequency.

Where:

Example Table:

Substitute the values:

Where:

Example Table: