Measure of Central Tendency in Statistics

 

Measures of Central Tendency

Topics:

what is statistics ? 

what is measure of central tendency.?

 1.Mean 

• Mean diagram image histogram
• Mean formula

2.Median

•   Median definition
•   Median meaning
•   Median Formula
•   Median calculate

3.Mode

•   Mode definition
•   Mode meaning
•   Mode Formula
•   Mode calculate 

What is statistics?

Statistics (data) is defined as the science of figure. It is the science of collecting, organizing, presenting, analyzing and interpretation the data.

Sir Ronald Fisher is the father of statistics.

          Graphical report                                                                   Hypothesis Testing

          Measure of central tendency                                              Correlation

          Measure of variability (Dispersion)                                     Regression etc

          Measure of position

          Skewness & kurtosis

Descriptive analysis statistics: We simply summarize the result from gathering data from a body, data group, or population and reaching conclusions about that group.

Inferential Statistics: We draw  inferences about the population from the sample are generated from the process of gathering simple data from a group ,body, population and reaching conclusion about the larger group from which the sample was drawn.

Population: - A population is the total collection of objects that are   of interest in a statistical study.​

Sample: - A sample, being a subset, is typically smaller than the population. In a statistical study, all elements of a sample are available for observation, which is not typically the case for a population.

 

Qualitative Data:- Qualitative data are measurements for which there is no natural numerical scale, but which consist of attributes, label or other non numerical characteristics.

 

Quantitative Data:- Quantitative data are numerical measurements that arise from a natural numerical scale.

    

Levels of Measurements

There are four different scales of measurement. The data can be defined as being one of the four scales. The four types of scales are:

 

• N-Nominal Scale
• O-Ordinal scale
• I- Interval scale
• R- Ratio Scale 

Level of measurement can be easily understood by the below figure.

Nominal Scale

A nominal scale is the 1st level of measurement scale in which the numbers serve as “tags” or “labels” to classify or identify the objects. A nominal scale usually deals with the non-numeric variables or the numbers that do not have any value. It is qualitative

 

Example:

An example of a nominal scale measurement is given below:

M- Male

F- Female

 

Ordinal Scale

The ordinal scale is the second level of measurement that reports the ordering and ranking of data without establishing the degree of variation between them. Ordinal represents the “order.” Ordinal data is known as qualitative data or categorical data. It can be grouped, named and also ranked.

Example:

Ranking of school students – 1st, 2nd, 3rd, etc.

Ratings in restaurants

 

Assessing the degree of agreement

Totally agree

Agree

Neutral

Disagree

Totally disagree

 

Interval Scale

The interval scale is the third level of measurement scale. It is defined as a quantitative measurement scale in which the difference between the two variables is meaningful. In other words, the variables are measured in an exact manner, not as in a relative way in which the presence of zero is arbitrary.

 

Example:

Likert Scale

Net Promoter Score (NPS)

Bipolar Matrix Table

 

Ratio Scale

The ratio scale is the fourth level of measurement scale, which is quantitative. It is a type of variable measurement scale. It allows researchers to compare the differences or intervals. The ratio scale has a unique feature. It possesses the character of the origin or zero points.

Example:

What is your weight in Kgs?

Less than 55 kgs

55 – 75 kgs

76 – 85 kgs

86 – 95 kgs

More than 95 kgs

 

What is measure of  central tendency?

 A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. It sometimes called measures of central location. The mean (which is also called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode. 

The mean, median and mode are all valid measures of central tendency, but we can use under different conditions, some measures of central tendency become more appropriate to use than others. 

       

 Mean

It represents the average value of the dataset. It can be calculated as the sum of all the values in the dataset divided by the number of values. In general, it is considered as the arithmetic mean. Some other measures of mean used to find the central tendency are as follows:

1) Geometric Mean

2) Harmonic Mean

3) Weighted Mean

It is observed that if all the values in the dataset are the same, then all geometric, arithmetic and harmonic mean values are the same. If there is variability in the data, then the mean value differs. Calculating the mean value is completely easy. The formula to calculate the mean value is given as

X1+X2+X3+X4+X5/N

• Individual Series

Example 1:2,3,4,5,6

Sol:

x¯¯¯=∑xN

 

=2+3+4+5+6/5

 =20/5

 =4   

Example:2-Find the mean from individual series.

20,25,30,15,20,40,60,50,

Sol-

 x¯¯¯=∑xN

A.M=x1+x2+x3+x4+x5+x6+x7+x8/N

20+25+30+15+20+40+60+54/8

=264/8

=33

• Discrete series

Example:3- Calculate mean from discrete series

x	f
5	3
10	7
15	5
20	12
25	9

Sol:-

Firstly we will calculate the values of Σfx

x	f	x*f
5	3	15
10	7	70
15	5	75
20	12	240
25	9	225
75	36	625

After calculating all values,

Σf=36 

n=5

Σfx=625

we will put all the values in the given formula.

x bar=  Σfx/Σf

=625/36

=17.36

Example:4-calculate mean from given series.

x	f
5	2
2	2
9	1
10	5
15	3
41	13

Sol:-

x bar=  Σfx/Σf

x	f	fx
5	2	10
2	2	4
9	1	9
10	5	50
15	3	45
41	13	118

x bar= Σfx/Σf

Σfx=118

Σf=13

Put the values in the formula.

=118/13

=9.07

Mean is 9.07.

Continuous Series:-

Example:5-

Calculate mean from given continuous series.

class	f
10-20	4
20-30	10
30-40	26
40-50	8
50-60	2

Sol.

In continuous series, first of all we need to calculate mid value(x).After that we follow the follow the same formula.

Class	Mid value (x)	f	fx
10-20	15	4	60
20-30	25	10	250
30-40	35	26	910
40-50	45	8	360
50-60	55	2	110
		50	1690

Put the values in the formula

x bar= Σfx/Σf

Σfx=1690

Σf=50

1690/50

=33.8

Mean is 33.8

Median

Median is the middle value of the dataset in which the dataset is arranged in the ascending order or in descending order. When the dataset contains an even number of values, then the median value of the dataset can be found by taking the mean of the middle two values.

• Individual series.

Median=

Example:1

Find the median for the following data set:

102, 56, 34, 99, 89, 101, 10, 54.
Sol-
Step 1: Place the data in ascending order (smallest to highest).
10, 34, 54, 56, 89, 99, 101, 102.

Step 2: Find the TWO numbers in the middle (where there are an equal number of data points above and below the two middle numbers).
10, 34, 54, 56, 89, 99, 101, 102

Step 3: Add the two middle numbers and then divide by two, to get the average:

   56 + 89 = 145

   145 / 2 = 72.5.

Discrete Series

Example:3 Calculate median in given discrete series.

x	f
5	2
2	2
9	1
10	5
15	3
41	13

Sol-

Median= n/2

=13/2

7.5

Now we will see where 7.5 lie in c.f  values

x	f	c.f
5	1	1
2	2	3
9	2	5
10	5	10
15	3	13
41	13

As we can see 7.5 lie where c.f  value is 10.Now we will see the value of x in front of C.F value, which is also 10.

So Median is 10

Continuous series

Example:4

Calculate Median in below series.

Class	f
10-20	4
20-30	10
30-40	26
40-50	8
50-60	2

Sol:-

First of all we will calculate mid value(x) and after that c.f.

Median=n/2

50/2

=25

Class	Mid value(x)	f	c.f
10-20	15	4	4
20-30	25	10	14
30-40	35	26	40
40-50	45	8	48
50-60	55	2	50
		50

For grouped data

(l+(n/2-c.f)/f*i

25 is lies in 40 in cf values

so l=30

n=50

c.f=14

f=26

By putting all the values

(30+(25-14)/26)*10

=30+.423*10

34.23

Mode

The mode represents the frequently occurring value in the dataset. Sometimes the dataset may contain multiple modes and in some cases, it does not contain any mode at all.

Individual Series

Example:1 

Find mode in given data set.

5, 4, 2, 3, 2, 1, 5, 4, 5    

Consider the given dataset 5, 4, 2, 3, 2, 1, 5, 4, 5                              

Since the mode represents the most common value. Hence, the most frequently repeated value in the given dataset is 5.

Discrete Series.

Example:2 Find the mode of given series.

x	f
5	2
2	2
9	1
10	5
15	3

Sol: The Arithmetic Mode of the given numbers is 10 as the highest frequency,5 is associated with 10.

Continuous Series.

Example:3 Calculate mode in continuous series.

Class	f
10-20	2
20-30	2
30-40	1
40-50	5
50-60	3

 

Sol-

Mode for grouped data  

Mo
=L+f1−f02f1−f0−f2×i

• 
  

• f$1_{}$ = Frequency of modal class

  ·        $f_0$ = Frequency of pre-modal class

  $f_2$ = Frequency of class succeeding modal class

      
  $i$ = Class interval.

•      =40+(5-1/2*5-1-3)*10

•      =46.6

Example:4 Calculate mode in given series.

Wages (in Rs.)	No. of workers
0-5	3
05-10	7
10-15	15-f0
15-20	30-f1
20-25	20-f2
25-30	10
30-35	5

Sol- Using following formula

=L+f1−f02f1−f0−f2×i

• 
  

• f0=  Frequency of class preceding modal class

• f$1$ = Frequency of modal class

  ·       $f_0$ = Frequency of pre-modal class

  $f_2$ = Frequency of class succeeding modal class

      
  $i$ = Class interval.

Substituting the values, we get

Mo=15+(30−15)/(2×30−15−20)×5

=15+3

=18

Relationship between mean, median and mode.

Mixed Questions

Ques.1 Find the missing frequency.

 A.M=17 yrs.

x (age)	f(no of persons)
8	3
20	2
26	?
29	1

Sol-

x (age)	f(no of persons)	fx
8	3	24
20	2	40
26	a=1	26a
29	1	29
	6+a	93+26a

 x̅ = Σfx/Σf

=93+26a/6+a

x̅ =17 given

=17(6+a)=93+26a

=102+17a=93+26a

=26a-17a=102-93

9a=9

a=1

A.M > G.M > H.M

Polled or combined Mean

A combined mean is a mean of two or more separate groups, 

Polled or combined mean=

X12bar     =N1 x̅1+ N2 x̅2/N1+N2

X123 bar       = N1 x̅1+ N2x2+ N3 x̅3/N1+N2+ N3 

Ques.2 

N1=8    x̅1=70 kg

N2=7    x̅2 =55 kg

N3=5    x̅3=71 kg

N1+N2+N3=20

X123 bar=?

Sol. 

X123 bar=  N1 x̅1+ N2x2+ N3 x̅3/N1+N2+ N3

=8*70+7*55+5*71/20

=560+385+355/20

=1300/20

=65kg

Ques.3- Given

N1+N2=Total no of student=150

N1=No of boys=?

N2= No of girls=?

x1 bar=70 kg

x2 bar=50 kg

x12 bar=60

x̅12= N1 x̅1+ N2x2/N1+N2

=70 N1+50N2/150

=60*150=70 N1+50 N2/150

=70 N1+50 N2=900

=7 N1+5 N2=900

=-5N1 -5N2=150*5=750

by cancelling 5n2

=2N1=150

=2N1=150

=N1=75

N1+N2=150

N2=150-75

N2=75

Ques.4 

X bar rainfall mon-sat=10 cm

Sunday = heavy rainfall x bar=15

A.M = x̅ =10 cm,

x̅  =  Σx/n

ΣX/6

Σx= nx̅ 

=6*10

=60

Total rainfall from mon to sat=60 cm

Mon to sun x̅ =15

x̅ =Σ/n

Σ/7

15= Σx/7

Σx=15*7=105 cm

Total rainfall from mon to sun=105 cm

 Rainfall on Sunday= Mon to sun-Mon -sat

=105-60

=45cm

Ques.5

x̅ =Σx/n

Σx=n x̅ =25*52=1300

Σx=1300 (incorrect)

Correct Σx=Incorrect Σx-wrong obs.+correct obs

=1300-64+44

=1280

Σx=1280

Correct x= Σx/n=1280/25=51.2

Based on the properties of the data, the measures of central tendency are selected.

• All three measure of central tendency hold good incase you have symmetrical distribution of continuous data. But most of the time , the analyst uses the mean because it involves all the value in the distribution or dataset. 

•  If you have skewed distribution, the best measure of finding the central tendency is the median. 

•  If you have the original data, then both the median and mode are the best choice of measuring the central tendency. 

•  If you have categorical data, the mode is the best choice to find the central tendency.