Measures of central tendency

By
Dr. Raksha Singh
Principal
Shri Shankaracharya Mahavidyalaya,
Junwani,Bhilai,C.G
Dr. Raksha Singh,15 February 2019
rakshasingh20@hotmail.com

Statistics
 Statistics is the process of organising raw
data(unorganised)in organised way to understand
information
 Statistics is important for understanding, describing
and predicting the world around you.
 Descriptive statistics summaries present information
that you have found . Summaries can be graphs or
small groups of numbers that are easier to understand
than long lists of numbers.
 Inferential statistics is using data to make predictions.
Both inferential statistics and descriptive statistics
help you understand the world around you and
communicate it effectively.

Measures of
Central Tendency
 Central tendency is a loosely defined concept
that has to do with the location of the center of
a distribution.
 Represent or describe large group of data with
a single number
 When working on a given set of data, it is not
possible to remember all the values in that set.
But we require an inference of the data given
to us. This problem is solved by mean median
and mode.

Introduction……
 Three measures of Central Tendency/Central
location :Mean, Median and Mode
 Sets of data show a distinct tendency to group
or cluster around a central point
 For any particular set of data, a single typical
value can be used to describe the entire set of
data
 Measures of central tendency are the center
values of a data set.

Introduction……
 Mean is the average of all the data. Its symbol
is x¯.
 Median is the middle value of the data set,
arranged in ascending order.
 Mode is the data value appearing most often in
the data set.

Mean
 The mean is the preferred measure of central
tendency because it considers all of the values in
the data set.
 However, the mean is not without limitations. In
order to calculate the mean, data must be
numerical.
 You cannot use the mean when you are working
with nominal data, which is data on characteristics
like gender, appearance, and race. For example,
there is no way that you can calculate the mean of
the girls' eye colors.

Which Measure Is Best?
While the mean, mode, and median represent centers of
data, one is usually more beneficial than another when
describing a particular data set.
For example, if the data has a wide range, the median is a
better choice to describe the center than the mean.
The income of a population is described using the median,
because there are very low and very high incomes in one
given region.
If the data were categorical, meaning it can be separated
into different categories, the mode may be a better choice.
If a sandwich shop sold ten different sandwiches, the mode
would be useful to describe the favorite sandwich.

How it is useful
 Central tendency is also useful when you want to compare
one piece of data to the entire data set.
 Let's say you received a 59% on your last statistics test,
which is usually in the D range.
 You go around and talk to your classmates and find out
that the average score on the test was 45%.
 In this instance, your score was significantly higher than
those of your classmates. Since your teacher grades on a
curve, your 59% becomes an A grade.
 Had you not known about the measures of central
tendency, you probably would have been really upset by
your grade and assumed that you performed badly/poorly
in the test.

What should you use?
 A.Data Set: 1,1,2,2,2,3,3,4,5,5
 Mean
 Median
 Mode
 B.Modified Data 1,1,2,2,2,3,3,4,5,5,600
 Mean-
 Median
 Mode

Solution
 A.
 Mean =2.8
 Median=2.5
 Mode=2
 B
 Mean =57.09
 Median=3
 Mode=2
In situation with outliers mean is
not a good measure instead of it
median and mode will be useful
depending upon the type of
information we are dealing with

Example
a) There were 29 books on the first shelf, 41 books on
the second shelf, and 23 books on the third shelf. Mary
rearranged the books so that there were the same
number of books on each shelf. After Mary rearranged
the books, how many were on the first shelf?
b) On an exam, two students scored 60, five students
scored 90, four students scored 75, and two students
scored 81. If the answer is 90, what is being asked in
the question (mean, median, mode, or range)?

When not to use the Mean
 Mean of the following is 30.7k,whereas salary of
workers revolves around 12 to 18k range. A.M skewed
by two outliers 90k & 95k
Staff 1 2 3 4 5 6 7 8 9 10
Salary 15k 18k 16k 14k 15k 15k 12k 17k 90k 95k

Median
 The Median is always in the middle. The median
is the value that cuts the data set in half.
 Arrange the data either in ascending or
descending order
 Outlier &skewed data have smaller effect on median
 For Individual and discrete series
 Median=( N+1)/2 (Individual and discrete)
 Continuous=L1 +(M-c/f) *L2-L1 For M=N/2

Example
Values 1 2 3 4 5 6 7 8
Freq 4 6 4 4 3 2 1 1
Q- Median is 3. Data is skewed to right so
mean will be higher
Values 1 4 6 8 9 10 11 12
Freq 1 1 2 3 4 4 5 5
Q- Median is 10. Data is skewed to left so mean is pulled
to left

Puzzle
 Mr A wanted to join Swimming class
he enquired mean age it was 17 and
median age also 17. So he thought
that class will be perfect.
 When he went to class guard stopped
him and asked where is your baby
 Mr A Confused… WHY?

Solution
 Mean & Median - 17
 In this class no person belongs to age 17???
 If we add another 3. Median will be 3. Not
consider adult
 If we add another 31. Median will be 31. Not
consider kids
Age 1 2 3 31 32 33
Frequency 3 4 2 2 4 3

Mode
 Third type of average
 Mode- Number with Highest
frequency
 Useful in categorical Data
 Last example. Mode = age 2 and 32.
which represents both category data
 Mode=3 Median – 2 Mean

Dr. Raksha Singh,15 feb 2019
The Effect of Skew on the Mean and Median and Mode
Positive Skew Mode < Median < Mean Negative Skew Mode > Median > Mean

Best method for
Qualitative Vs Quantitative
 If the data being analyzed is
qualitative, then the only measure of
central tendency that can be reported
is the mode.
 However, if the data is quantitative
in nature (ordinal or interval/ratio)
then the mode, median, or mean
can be used to describe the data.

Example
https://www.youtube.com/watch?v=5C9LBF3b65s

Thank You

Measures of central tendency

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