Maharashtra Board Class 8 Maths part 2 Chapter 11 Statistics PDF Download

Statistics

Let's Recall

Ex. The numbers of pages of a book Ninad read for five consecutive days were 60, 50, 54, 46, 50. Find the average number of pages he read everyday.

Solution: Average = \(\frac{\text{Sum of all scores}}{\text{Total number of scores}}\)

\(= \frac{60 + 50 + 54 + 46 + 50}{5} = \frac{260}{5} = 52\)

Therefore, the average number of pages read daily is 52.

The average is also known as 'Arithmetic Mean' or 'Mean'.

Teacher's Note

Ninad read pages over five days. We add all pages and divide by 5 days to find the average. This is like sharing sweets equally among children - everyone gets the same amount.

Exam Trick

Remember: Average means equal sharing. Add all numbers, then divide by how many numbers there are. Like dividing chocolates equally among friends.

Points to Remember

Average = Sum of all numbers ÷ Count of numbers
Average is also called Arithmetic Mean or Mean
We use average to understand data quickly
More numbers give a better average
Average helps us compare different groups

Let's Learn

The number of pages read everyday in the above example is numerical data. The conclusion that Ninad read about 50 pages everyday is drawn from the numerical data.

In this way, collecting information regarding certain problem or situation, analysing this information and after interpretation drawing conclusion about it, is a separate branch of knowledge. This branch is known as 'Statistics'.

Arithmetic Mean

We have seen that the average of 60, 50, 54, 46 and 50 is 52. The average is called as 'arithmetic mean' or simply 'mean' in statistical language. To find the mean of numerical data, we add all the scores in the data and divide the sum by total number of scores. Let's study some aspects of the method. See the following example.

Ex. The marks obtained by 37 students of standard 8 in a test of 10 marks are given below. Find the mean of the data.

2, 4, 4, 8, 6, 7, 3, 8, 9, 10, 10, 8, 9, 7, 6, 5, 4, 6, 7, 8, 4, 8, 9, 7, 6, 5, 10, 9, 7, 9, 10, 9, 6, 9, 9, 4, 7.

Teacher's Note

Class 8 students took a test of 10 marks. We count how many got each mark, then use a table to find the average mark. This is like counting how many students scored 4 marks, how many got 5 marks, and so on.

Exam Trick

Remember: Use a table with tally marks to count. Never add all numbers one by one - it takes too long. Group same numbers together like sorting marbels into colors.

Points to Remember

Tally marks help count quickly - five marks like //// with a line means 5
Frequency means how many times a number appears
Multiply each number by its frequency first
Add all these products together
Divide by total count to get the mean

Solution: If we go on adding the scores one after the other, it will take lot of time. We know that 7 + 7 + 7 + 7 + 7 = 7 × 5 = 35. This simplifies the operation of addition, when a number is to be added to itself. So let us use the method and add the numbers by classifying them.

Mean = \(\frac{\sum f_i \times x_i}{N} = \frac{259}{37} = 7\)

Prepare a table as above and follow the steps to calculate the mean of a data.

Write the scores in the 1st column, in ascending order as \(x_1 < x_2 < x_3\) ....

Write the tally marks in next column.

Count the tally marks of scores and write the frequency of the score, denoted as \(f_i\). Write the sum of all frequencies below the frequency column. The total frequencies is denoted by 'N'.

In last column write the products \(f_i \times x_i\). The sum of \(f_i \times x_i\) is denoted as \(\sum f_i \times x_i\). '\(\sum\)' (sigma) indicates the 'sum'. Arithmetic mean is denoted by \(\bar{x}\)

Mean \(\bar{x} = \frac{\sum f_i \times x_i}{N}\)

Teacher's Note

We make a table with tally marks to count. Then multiply each mark by how many students got that mark. This method is fast and clear, like organizing students by height in rows.

Exam Trick

Remember: The formula is \(\bar{x} = \frac{\sum f_i \times x_i}{N}\). Write \(\sum\) which means "add all." Always multiply first, then add, then divide - never divide before multiplying.

Points to Remember

\(\bar{x}\) is the symbol for mean
\(x_i\) means each different score
\(f_i\) means frequency or count
N is total number of items
\(\sum\) means add everything together

This is a preview of the first 3 pages. To get the complete book, click below.