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Statistics

On completing this lesson students will know the answers for the following, problem solving methods.

1. What is meant by analysis of Data?

2. What is involved in the interpretation of Data?

3. Any data collected is useless if it is not properly analyzed and interpreted.

4. Methods to find arithmetic mean Median, Mode, and Graph of data.

Measures of central tendency. This is a statistical answer for a value among data, that shows all data is centered around a particular value which is called data at central tendency. Mean, Median, Mode, Geometrical and Harmonic means are different measures of Central tendency.

Mean Arithmetic Mean is the simplest measure.

If x₁, x₂, x₃ … x_n are n data (observations) then( x₁+ x₂+ x₃+ …+ x_n )/n is called Arithmetic Mean dented by

Example: Find the mean of 21, 26, 28, 35, 32

Here n = 5 and the mean = = 28.4

Arithmetic mean of grouped data

A grouped data consists of data and its frequency. The following are two methods to compute Arithmetic mean.

(1) The direct Method (2) The short – cut method.

The direct method:

Method:

1. Write the values of the variable in the first column and the corresponding frequencies in the second column.
2. In the third column find fx
3. Find the sum of all entries in the third column. i. e. åfx.
4. Calculate the sum of all frequencies åf
5. Use the formula Mean =

The following three columns are formed, åf, åfx are found. By using the formula =we get Arithmetic mean.

Example: Find the mean of the following data

x	10	12	20	25	35
f	3	10	15	7	5

Answer:

X	F	fx
10	3	30
12	10	120
20	15	300
25	7	175
35	5	175
Total	åf=40	åfx=800

Therefore Mean = = = 20.

Short Cut Method:

Method:

6. Write the values of the variable in the first column and the corresponding frequencies in the second column.
1. If the class interval is given find mid values and use it in column 1. Formula to find mid value is ½(upper limit +lower limit of a class).
7. From the first column, choose a number ‘A’ as the ‘assumed mean’ . Calculate d= x – A and write it against the corresponding frequencies in the third column.
8. In the fourth column calculate fd.
9. Find the sum of all entries in the fourth column. i. e. åfd.
10. Calculate the sum of all frequencies åf
11. Use the formula Mean =A+

USAGE of step deviation method: It is useful when data values are large.

Example: Find the mean of the following distribution.

Class	0 -10	10 – 20	20 – 30	30 – 40	40 – 50
Frequency	6	8	12	8	6

Applying the above mentioned methods we create the following columns and do all steps.

Class interval

Mid Value

Frequency

D = x – A

Fd

Class interval	Mid Value X	Frequency f	D = x – A	Fd
0 – 10	5	6	-20	-120
10 – 20	15	8	-10	-80
20 – 30	A=25	12	0	0
30 – 40	35	8	10	80
40 – 50	45	6	20	120
		åf=40		åfd=0

Using the formula Mean =A+ = 25 + 0/40 = 25. ANSWER: Mean = 25

Now that you have learnt to find MEAN here are practice problems.

Median

Median is another measure of central tendency. It is that value which divides the data into two equal parts.

How to find Median?

1. If data is ungrouped, arrange them in ascending or descending order.
2. Find the middle value using
1. If number of data is odd , the (n+1)^th/2 datum
2. If the number of data is even then it is the average of n/2^th and (n+1)^th/2 observation.

Example 1: The median of 7, 18, 9, 8, 12 is calculated thus

Arranging data in ascending order 7, 8, 9, 12, 18.

There are 5 data. n =5.

N is odd the median is (n+1) ^th/2 datum = (5+1)^th/2 = 3 ^rd observation.

Therefore Median is 9.

Example 2 : Find the median of 6, 4, 8, 12, 9, 11

Arrange them in ascending order. 4, 6, 8, 9, 11, 12. There are 6 data. N=6.

The median is average of n/2^nd = 3^rd and (n+1)^th/2 = 4^th Observation.

Median = (8 + 9) /2

= 8.5.

II when the data is grouped.

1. Prepare the less-than cumulative frequency distribution

2. Find N/2

3. Find the cumulative frequency just greater than N/2.

4. The corresponding value of the variable x gives the median.

Example 3: Find the median of

X	3	4	5	7	9	11
F	2	5	13	7	3	1

Answer:

x	f	Cf
3	2	2
4	5	7
5	13	20
7	7	27
9	3	30
11	1	31
	31

N = 31 which is odd.

The middle value = [ (31 + 1)/2]^th value = 16^th .

After the 7^th observation, up to the 20^th observation the value of x is 5.

Therefore the median is 5.

III. Median of a continuous frequency distribution (grouped data)

1. Calculate the less-than cumulative frequency for each class.

2. Find N/2

3. From the cumulative frequencies, locate the median class where the median value of (N/2)th item falls.

4. Let l be the lower limit of the median class.

5. Let f be the frequency of the median class.

6. let c be the width of the class interval.

7. Let m be the cumulative frequency of the class just before the median class.

8. The median = l +

Example:

Find the median for the following distribution

Class Interval	5 – 12	12 – 19	19 – 26	26 – 33	33 – 40	40 – 47
Frequency	3	6	12	16	8	5

Solution:

Class	Frequency	CF
5 – 12	3	3
12-19	6	9
19 – 26	12	21
26 – 33	16	37
33 – 40	8	45
40 – 47	8	50
	50

N/2 = 50/2 = 25

The Cumulative frequency just greater than 25 is 37. The corresponding class 26 – 33 is the Median Class.

L = 26, m =21, c=7 , N/2 =25, f =16. Substituting in the formula we get

Median = l +

= 26 +

= 26 + 28/16

= 26 + 1.75

= 27.75

Now that you have learnt to find MEDIAN here are practice problems

Mode:

Mode is that value of the variable in the data which occurs most frequently. The data which has maximum frequency is called Mode.

Mode is the most useful measure of central tendency used in trade and industry.

The sizes of shoes, readymade garments, are decided by Mode.

Method to find Mode:

1. If the data is ungrouped (discrete) then locate the observation which has maximum frequency. This observed data is Mode.
2. If the data is grouped the following formula is used:

Mode = l +

Where l is the Lower limit of the modal class (class with maximum frequency)

f_m = Frequency of the model class

f₁ = Frequency of class preceding the modal class.

f₂ = Frequency of the class succeeding the modal class

h = width of the modal class.

Example:

The following table gives the length of life of 160 bulbs. Find the mode.

Life (in hours)	No. of Bulbs
0 – 400	5
400 – 800	16
800 – 1200	40
1200 – 1600	41
1600 – 2000	27
2000 – 2400	15
2400 – 2800	10
2800 – 3200	6

Maximum frequency is 41. The class which has maximum frequency is 1200 – 1600.

This is the modal class.

Where l = 1200

f_m = Frequency of the model class = 41

f₁ = Frequency of class preceding the modal class. = 40

f₂ = Frequency of the class succeeding the modal class = 27

h = width of the modal class. = 400

Mode = l += 1200 +

= 1200 + 400/15

= 1200 + 26.666

= 1226.67 Hours.

. Now that you have learnt to find MODE here are practice problems

This tutorial is developed with an intention of learning, testing and know your level of learning

© Success systems, Chennai, India.