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Revision Notes on Statistics

Statistics

Statistics is one of the parts of mathematics in which we study about the collecting, organizing, analyzing, interpreting and presenting data.

Statistics is very helpful in real life situations as it is easy to understand if we represent a data in a particular number which represents all numbers. This number is called the measure of central tendency. Some of the central tendencies commonly in use are -

Mean

It is the average of “n” numbers, which is calculated by dividing the sum of all the numbers by n.

The meanof n values x1, x2, x3, ...... xn is given by

Mean

Median

If we arrange the numbers in an ascending or descending order then the middle number of the series will be median. If the number of series is even then the median will be the average of two middle numbers.

If n is odd then the median isOddobservation.

If the n is even then the median is the average ofEvenobservation.

Mode

The number which appears most frequently in the series then it is said to be the mode of n numbers.

Mean of Grouped Data (Without Class Interval)

If the data is organized in such a way that there is no class interval then we can calculate the mean by

Mean of Grouped Data

where, x1, x2, x3,...... xn are the observations

f1, f2, f3, ...... fn are the respective frequencies of the given observations.

Example

Grouped Population Mean
x f fx
20 40 800
40 60 2400
60 30 1800
80 50 4000
100 20 2000
  200 ∑fx = 11000 

Here, x1, x2, x3, x4, x5 are 20, 40, 60, 80, 100 respectively and f1, f2, f3 , f4, f5 are 40, 60, 30, 50, 20 respectively.


means

Mean of Grouped Data (With Class-Interval)

When the data is grouped in the form of class interval then the mean can be calculated by three methods.

1. Direct Method

In this method, we use a midpoint which represents the whole class. It is called the class mark. It is the average of the upper limit and the lower limit.

Formula of class mark

Direct Method

Example

A teacher marks the test result of the class of 55 students for mathematics. Find the mean for the given group. 

Marks of Students 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
Frequency 27 10 7 5 4 2

To find the mean we need to find the mid-point or class mark for each class interval which will be the x and then by multiplying frequency and midpoint we get fx.

Marks of students Frequency(f) Midpoint(x) fx
0 – 10 27 5 135
10 – 20 10 15 150
20 – 30 7 25 175
30 – 40 5 35 175
40 – 50 4 45 180
50 – 60 2 55 110
    ∑f = 55 ∑fx = 925

frequency

2. Deviation or Assumed Mean Method

If we have to calculate the large numbers then we can use this method to make our calculations easy. In this method, we choose one of the x’s as assumed mean and let it as “a”. Then we find the deviation which is the difference of assumed mean and each of the x. The rest of the method is the same as the direct method.

Deviation or Assumed Mean Method

Example

If we have the table of the expenditure of the company's workers in the household, then what will be the mean of their expenses?

Expense(Rs.) 100 - 150 150 - 200 200 - 250 250 - 300 300 - 350 350 - 400
Frequency 24 40 33 28 30 22

Solution

As we can see that there are big values of x to calculate so we will use the assumed mean method.

Here we take 275 as the assumed mean.

Expenses(Rs.) Frequency(f) Mid value(x) d = x – 275 fd
100 – 150 24 125 - 150 - 3600
150 – 200 40 175 - 100 - 4000
200 – 250 36 225 - 50 -1650
250 – 300 28 275 0 0
300 – 350 30 325 50 1500
350 – 400 22 375 100 2200
  ∑f = 180     ∑fd = - 5550

method

3. Step Deviation Method

In this method, we divide the values of d with a number "h" to make our calculations easier.

Step Deviation Method

Example

The wages of the workers are given in the table. Find the mean by step deviation method.

Wages  20 - 30 20 - 30 30 - 40 40 - 50 50 - 60
No. of workers 8 9 12 11 6

Solution

Wages No. of workers (f)  Mid-point(x) Assume mean (a) = 35, d = x - a h = 10, u = (x – a)/h fu
10 – 20 8 15 -20 -2 -16
20 – 30 9 25 -10 -1 -9
30 – 40 12 35 0 0 0
40 – 50 11 45 10 1 11
50 – 60 6 55 20 2 12
  ∑f = 46       ∑fu = -2

deviation

Mode of Grouped Data

In the ungrouped data the most frequently occurring no. is the mode of the sequence, but in the grouped data we can find the class interval only which has the maximum frequency number i.e. the modal class.

The value of mode in that modal class is calculated by

Formula of mode

l = lower class limit of the modal class

h = class interval size

f1 =frequency of the modal class

f0 =frequency of the preceding class

f2 = frequency of the succeeding class

Example

The table of the marks of the students of a class is given. Find the modal class and the mode.

Marks 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100
No. of students 4 8 6 7 5

Solution

Here we can see that the class interval with the highest frequency 8 is 20 – 40.

So this is our modal class.

Modal class = 20 - 40

Lower limit of modal class (l) = 20

Class interval size (h) = 20

Frequency of the modal class(f1) = 8

Frequency of the preceding class(f0) = 4

Frequency of the succeeding class (f2) = 6

Mode

Median of Grouped Data

To find the median of a grouped data, we need to find the cumulative frequency and n/2

Then we have to find the median class, which is the class of the cumulative frequency near or greater than the value of n/2.

Cumulative Frequency is calculated by adding the frequencies of all the classes preceding the given class.

Then substitute the values in the formula

Formula of median

where l = lower limit of median class

n = no. of observations

cf = cumulative frequency of the class preceding to the median class

f = frequency of the median class

h = size of class

Example

Find the median of the given table.

Class Interval Frequency Cumulative Frequency (fc)  
1 – 5 4 4 4
6 – 10 3 7 4 + 3 = 7
11 – 15 6 13 7 + 6 = 13
16 – 20 5 18 13 + 5 = 18
21 – 25 2 20 18 + 2 = 20 
  N = 20    

Solution

Let’s find the n/2.

n = 20, so n/2 = 20/2 = 10

The median class is 11 - 15 as its cumulative frequency is 13 which is greater than 10.

Median

13.5

Remark: The empirical relation between the three measures of central tendency is

3 Median = Mode + 2 Mean

Graphical Representation of Cumulative Frequency Distribution

The graph makes the data easy to understand. So to make the graph of the cumulative frequency distribution, we need to find the cumulative frequency of the given table. Then we can plot the points on the graph.

The cumulative frequency distribution can be of two types -

1. Less than ogive

To draw the graph of less than ogive we take the lower limits of the class interval and mark the respective less than frequency. Then join the dots by a smooth curve.

2. More than ogive

To draw the graph of more than ogive we take the upper limits of the class interval on the x-axis and mark the respective more than frequency. Then join the dots.

Example

Draw the cumulative frequency distribution curve for the following table.

Marks of students 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
No. of students 7 10 14 20 6 3

Solution

To draw the less than and more than ogive, we need to find the less than cumulative frequency and more than cumulative frequency.

Marks No. of students Less than cumulative frequency More than cumulative frequency
0 – 10 7 Less than 10 7 More than 0 60
10 – 20 10 Less than 20 17 More than 10 53
20 – 30 14 Less than 30 31 More than 20 43
30 – 40 20 Less than 40 51 More than 30 29
40 – 50 6 Less than 50 57 More than 40 9
50 – 60 3 Less than 60 60 More than 50 3
        More than 60 0

Now we plot all the points on the graph and we get two curves.

The less than and more than ogive graph

Remark

  • The class interval should be continuous to make the ogive curve.

  • The x-coordinate at the intersection of the less than and more than ogive is the median of the given data.


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