measures of central tendency
TRANSCRIPT
Measures of central tendency
• A measure of central tendency is a typical value around which other figures concentrate.
• An average is a single value within the range of the data that is used to represent all the values in the series.
Objectives of an average
• It reduces the mass information into a single value.
• Average provides facility to make comparisons.
• Average is very useful in statistical analysis.• Average is also used in sampling. We can get a
clear picture of a group or population by the help of a sample.
Arithmetic Mean
• It is defined as the sum of the observations divided by the number of observations.
• It is denoted by x
1. Individual Series
(i) Direct Method
where, x denotes the value of the
variable n denotes the number of
observations.
(ii) Short Cut Method
where, A is assumed mean n denotes number of
terms.
xn
x1 Axdd
nAx ;
1
Ques 1. The number of new orders received by a company over the last 25 working days were recorded as follows:
3, 0, 1, 4, 4, 4, 2, 5, 3, 6, 4, 5, 1, 4, 3, 2, 0, 2, 0, 5, 4, 2, 3, 3, 1.
Calculate the average number of orders received over all similar working days.
Ques 2. The following data gives the equity holdings of 10 of the India’s billionaires.
Calculate the average equity holding.
Name Equity Holdings(millions of Rs)
Kiran Mazumdar-Shaw 2717
The Nilekani family 2796
The punj family 3098
Karsanbhai K. Patel 3144
Shashi Ruia 3527
K.K.Birla 3534
B.Ramalinga Raju 3862
Habil F. Khorakiwala 4187
The Murthy family 4310
Keshub Mahindra 4506
2. Discrete Series(i) Direct Method
where, x denotes the values of
the variable f denotes the frequency N = ∑f ; denotes the no.
of observations.
(ii) Short Cut Method
where, d = x - A A is assumed meanN = ∑f ; denotes the no. of
terms
xfN
x1
dfN
Ax1
Ques 1. The HR manager at a city hospital began a study of the overtime hours of the registered nurses. The following data was recorded during a month.
Calculate the arithmetic mean of overtime hours during the month.
Overtime hours
No. of nurses
5 3
6 4
7 2
9 4
10 2
12 5
13 3
15 2
Ques 2. The following are the figures of profits earned by 1,400 companies during 2008-2009.
Calculate the average profit for all the companies.
Profits(Rs. Lakhs)
No. of companies
300 500
500 300
700 280
900 120
1100 100
1300 80
1500 20
3. Continuous Series(i) Direct Method:
where, x denotes the mid point of various class interval f denotes the frequency of each class N = ∑f ; denotes the total frequency
xfN
x1
(ii) Short Cut Method:
where, d = x – A x denotes the mid point of various class interval A is assumed mean f denotes the frequency of each class N = ∑f ; denotes the total frequency
dfN
Ax1
(iii) Step Deviation Method:
where, u = (x - A)/h h denotes the width of class interval A is assumed mean N = ∑f ; denotes the no. of terms
hufN
Ax1
Ques 1. Calculate mean for the following frequency table: Weekly rent No. of persons
paying rent
200-400 6
400-600 9
600-800 11
800-1000 14
1000-1200 20
1200-1400 15
1400-1600 10
1600-1800 8
1800-2000 7
Ques 2. In an examination of 675 candidates the examiner supplied the following information:
Marks obtained
No. of candidates
0 – 10 7
10 – 20 32
20 – 30 56
30 – 40 106
40 – 50 175
50 – 60 164
60 – 70 86
70 – 80 44
Ques. Six types of workers are employed in each of two workshops but at different rates of wages as follows:
In which of the two workshops is the average rate of wages per worker higher and by how much?
Workshop ADaily wagesper worker
No. of workers
Workshop BDaily wages
No. of workersper worker
92.50 2 93.00 11
93.50 14 93.25 50
94.00 20 93.50 8
93.00 7 94.25 10
94.25 6 94.50 10
94.50 1 95.00 2
Ques. Calculate mean from the following data pertaining to the profits(in crore Rs.) of 125 companies: Profits(Rs. crore)(less than)
No. of companies
10 4
20 16
30 40
40 76
50 96
60 112
70 120
80 125
Merits & Demerits• Merits
– All values are used– It has unique value & easy to calculate – The sum of the deviations from the mean is 0
• Demerits– The mean is affected by extreme values
E.g., average salary at a company 12,000; 12,000; 12,000; 12,000; 12,000; 12,000;
12,000; 12,000; 12,000; 12,000; 20,000; 390,000 Mean = $44,167– It is not suitable for open end classes
MEDIAN
The median is the measure of central tendency which appears in the middle of an ordered sequence of values. That is, half of the observations in a set of data are lower than it and half of the observations are greater than it.
1. Individual Series
• Arrange the data in increasing or decreasing order.
• Median is given by:
Median= termthn
2
)1(
Questions:1.From the following data of wages of 7 workers, compute the median wage: 2600, 2650, 2580, 2690, 2660, 2606, 2640
2. Discrete Series
• Calculate the cumulative frequency.• Find (N+1)/2th term.• Select the cumulative frequency in which that
value lies.• The value of the variable corresponding to
that cumulative frequency is the median.
3. Continuous Series• Calculate the cumulative frequency.• The value corresponding to N/2th term gives the median class.• Then median is calculated by the formula:
ervalclassofwidthh
classmedianthe
preceedingclasstheoffrequencycumulativecf
classmediantheoffrequencyf
classmediantheofitlowerl
where
hcfN
flMedian
int
lim
,
2
1
3. Calculate the median for following data:
Class interval Frequency
10 - 15 11
15 – 20 20
20 – 25 35
25 – 30 20
30 – 35 8
35 - 40 6
4. Calculate the median from the following data pertaining to the profits (in crore Rs.) of 125 companies:
Profits (less than) No. of companies
10 4
20 16
30 40
40 76
50 96
60 112
70 120
80 125
• Merits – Median is unique– Median is less affected by extreme values as compared to
mean– It can be used for open–end distribution– Graphical presentation of median is possible– Median is used for studying qualitative attributes
• Demerits – For median, it is necessary to arrange the data– It is not capable for further algebraic treatment– It does not use each and every observation of the data set
Mode• The most frequent score in the distribution.
• A distribution where a single score is most frequent has one mode and is called unimodal.
• A distribution that consists of only one of each score has n modes.
• When there are ties for the most frequent score, the distribution is bimodal if two scores tie or multimodal if more than two scores tie.
1. Individual Series: In Individual series we guess mode by inspection. We observe that term in the
series which occurs maximum number of times. This term is called mode.
2. Discrete Series: In discrete series mode is that value of variable whose frequency is maximum.
3. Continuous Series:
The class interval that is corresponding to the maximum frequency is the modal class. Then mode is calculated by the formula:
classalofwidthh
classalthesuceedingclassoffrequencyf
classalpreceedingclassoffrequencyf
classaloffrequencyf
classalofitlowerl
where
hfff
fflz
mod
mod
mod
mod
modlim
,
2
2
0
1
201
01
Ques 1. Find the mode of following data:
Class Frequency
0 – 5 29
5 – 10 195
10 – 15 241
15 – 20 117
20 – 25 52
25 – 30 10
30 – 35 6
35 – 40 3
40 – 45 2
Ques 2. Find the mode of following data:
Class Frequency
0 – 10 5
10 – 20 15
20 – 30 20
30 – 40 20
40 – 50 32
50 – 60 14
60 – 70 14
70 – 80 5
Merits of Mode:
• It is easy to understand. Sometimes it is found only by inspection.• It is least affected by extreme values.• It can also be located graphically.• It is also useful when items related to fashion are considered.
Demerits of Mode:
• It is not based on all the observations.• Equal intervals are needed for the calculation of mode.
Empirical relation between mean, median and mode
The relationship between arithmetic mean, median and mode is given by following formula:
Mode = 3 Median – 2 Mean