Group efforts

1. Measures of Central Tendency
Presented to: Dr. Naila Alam
Presented by: Sumia Syed
Umme Habiba
Sobia Idrees
Sajda Aish

2. Presented by: Sumia Syed

4. Central Tendency:
Overview: In statistics, the
concept of an average, or
representative, score is called
Central Tendency.
 The goal in measuring central
tendency is to describe a
distribution of scores by
determining a single value that
identifies the center of the
distribution.

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• Ideally, this central value is the
score that is the best
representative value for all of
the individuals in the
distribution.

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• In everyday language, central
tendency attempts to identify the
‘average’ or ‘typical’ individual. This
average value can then be used to
provide a simple distribution of an
entire population or a sample.
• In addition, to describing an entire
distribution, measures of central
tendency are also useful for making
comparisons between groups of
individuals or between set of figures.

7. Methods: Measuring
Central Tendency:
• Statisticians have developed
three different methods for
measuring central tendency;
1.The Mean
2.The Median and
3.The Mode

8. • Values called measures of
central tendency are used to
summarize data into a single
value or statistic.
– The mean is the sum of all the data
values divided by the number of values.
– The median is the middle number when
the data are arranged in order.
– The mode is the value that occurs most
frequently in the data.
Continue..

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• 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. As such,
measures of central tendency
are sometimes called measures
of central location.

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• They are also classed as
summary statistics.
• The mean (often 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.

11. Examples of Measures
of Central Tendency:
• For the data 1,2,3,4,5,5,6,7,8
the measures of central
tendency are;
• Mean =
• Median = 5
• Mode = 5

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• The mean, median and mode are all
valid measures of central tendency,
but under different conditions, some
measures of central tendency
become more appropriate to use
than others.
• In the following sections, we will look
at the mean, median and mode, and
learn how to calculate them and
under what conditions they are most
appropriate to be used.

14. Mean
Presented by: Sobia idrees

15. M y topic Objectives are:
By the end of this presentation, you should
be able to:
To identify the types of central tendencies.
Compute the mean of a given set of data.
Applications of mean.

16. Mean
The mean is the sum of the value of each observation in
a dataset divided by the number of observations. This is
also known as the arithmetic average.
The mean, often called the average, of a numerical set of data,
is simply the sum of the data values divided by the number of
values. The mean is the balance point of a distribution. The
calculations for the mean of a sample and the total population
are done in the same way. However, the mean of a population
is constant, while the mean of a sample varies from sample to
sample

17. Continued
The mean of a sample or a population is computed by
adding all of the observations and dividing by the
number of observations.
Population mean = μ = ΣX / N OR Sample mean
= x = Σx / n
where ΣX is the sum of all the population observations,
N is the number of population observations, Σx is the
sum of all the sample observations, and n is the number
of sample observations.

18. Continued
When statisticians talk about the mean of a
population, they use the Greek letter μ to
refer to the mean score. When they talk
about the mean of a sample, statisticians
use the symbol x to refer to the mean score.

20. Group Data vs.Ungrouped Data
Group Data
Grouped data is when there is a large number of
possible outcomes, we will usually need to group the data.
For example: The ages of 200 people entering a park on a
Saturday afternoon. The ages have been grouped into the
classes 0-9, 10-19, 20-29, etc.

21. Arithmetic Mean (group-data) :
Formula:
Arithmetic Mean = ΣfX/Σf
where
X = Individual value
f = Frequency
The symbol of Σ is pronounced as sigma and is used to represent the sum of number.
Grouped Data Arithmetic Mean Example:
X -Value Frequency(f) ΣfX
1 2 1 * 2 = 2
2 3 2 * 3 = 6
3 2 3 * 2 = 6
Σf = 7 ΣfX= 14
Step 1: Find Σf.
Σf = 7
Step 2: Now, find ΣfX.
ΣfX = 14
Step 3: Now, Substitute in the above formula given
Arithmetic mean = ΣfX/Σf = 14/7 = 2

22. Ungrouped data
Ungrouped data is the opposite of grouped data
with only one possible answer.
For example: The ages of 200 people
entering a park on a Saturday afternoon. The
ages are:
27, 8, 10, 49 etc.

23. Arithmetic Mean (ungroup-data)
Formula:
Mean = sum of elements / number of
elements
= a1+a2+a3+.....+an/n
Arithmetic Mean = ΣX/n
where
X = Individual value
n = Total number of values

24. Example:
To find the mean of 3,5,7.
Step 1 Find the sum of the numbers.
3+5+7 = 15
Step 2: Calculate the total number.
there are 3 numbers.
Step 3: Finding mean.
15/3 = 5
Ans = 5

25. Shape of distribution
Symmetrical distributions:
When a distribution is symmetrical, the mode,
median and mean are all in the middle
of the distribution.

26. .
Skewed distributions:
When a distribution is skewed the mode
remains the most commonly occurring value,
the median remains the middle value in the
distribution, but the mean is generally ‘pulled’
in the direction of the tails. In a skewed
distribution, the median is often a preferred
measure of central tendency, as the mean is
not usually in the middle of the distribution.

27. Positive skewed distribution:
A distribution is said to be positively or right skewed
when the tail on the right side of the distribution is longer
than the left side. In a positively skewed distribution it is
common for the mean to be ‘pulled’ toward the right tail
of the distribution.
Positively mean
> median

28. Negative skewed distribution:
A distribution is said to be negatively or left
skewed when the tail on the left side of the
distribution is longer than the right side. In a
negatively skewed distribution, it is common for the
mean to be ‘pulled’ toward the left tail of the
distribution.
Negatively mean
< median

29. Advantages of Mean:
• It is easy to understand & simple calculate.
• It is based on all the values.
• It is easy to understand the arithmetic average even if
some of the details of the data are lacking.

30. Disadvantages of Mean:
• It is affected by extreme values.
• It cannot be calculated for open end classes.
• It cannot be located graphically
• It gives misleading conclusions.

31. Median
Presented by: Umm-e-Habiba

32. The Median
• The second measure of central
tendencies
• The goal of the median is to
locate the midpoint of the
distribution.
• There are no specific symbols
or notions to identify the
median

33. Conti.…
• The median is simply identified by
the word median.
• In addition, the definition and the
computations for the median are
identical for a sample and for a
population.

34. The Definition
• If the scores in a distribution
are listed in order from
smallest to largest
• The median is the midpoint of
the list
• More specifically, the median is
the point on the measurement
scale below which 50% of the
scores in the distribution are
located.

35. Finding the Median for
most distribution
• The scores are divided into
equal-sized group.
• We are not locating the
midpoint from highest to lowest
X values
• To find the median, list the
scores in order from smallest
to largest

36. Conti.…
• Begin with the smallest score
and count the score as you
move up the list
• The median is the first point you
reach that is greater than of
50% of the score in the
distribution
• The median can be equal to a
score in the list or it can be a
point between two scores

37. Conti.…
• Notice that the median is not
algebraically defined (there is
no equation for computing the
median)
• Means that there is a degree of
subjectivity in determining the
exact value

38. Example
• This example demonstrates the
calculation of the median when n is an
“odd” number.
3, 5, 8, 10, 11

39. Example
•This example demonstrates the calculation
of the median when n is an “even” number
1, 1, 4, 5, 7, 8

40. Example: Median of a set Grouped Data
in a Distribution of Respondents by age
Age Group Frequency of
Median class(f)
Cumulative
frequencies(cf)
0-20 15 15
20-40 32 47
40-60 54 101
60-80 30 131
80-100 19 150
Total 150

42. Merits
1. Simplicity
– It is very simple measure of central
tendency
1. Free from the effect of
extreme values
2. Real value
• Representative value as
compared to arithmetic mean
average, the value of which may
not exist in the series at all.

43. Cont.…
Graphic presentation
– It can be estimated also through the graphic
presentation of data.
Possible even when data is incomplete

44. Demerits
Unrealistic
– When the median is located somewhere
between the two middle values, it remains
only an approximate measure, not a precise
value.

45. Cont.…
Lack of representative character
– limited representative character as it is not
based on all the items in the series.

46. Cont.…
Lack of algebraic treatment
– Arithmetic mean is capable of
further algebraic treatment, but
median is not. For example,
multiplying the median with the
number of items in the series will
not give us the sum total of the
values of the series.

47. Mode
Presented by: Sajda Aish

48. Mode
 Mode is the most frequent value or
score in the distribution.
 It is defined as that value of the item in
a series.
 It is denoted by the capital letter Z.
 Highest point of the frequencies
distribution curve.

49. Croxton and Cowden : defined it as “the
mode of a distribution is the value at the
point armed with the item tend to most
heavily concentrated. It may be regarded
as the most typical of a series of value”
The exact value of mode can be obtained
by the following formula.
Z=L1
+

50. Monthly rent (Rs) Number of Libraries (f)
500-1000 5
1000-1500 10
1500-2000 8
2000-2500 16
2500-3000 14
3000 & Above 12
Total 65
Example: Calculate Mode for the distribution of
monthly rent Paid by Libraries in Karnataka

51. Z=2000+
Z =2000+
Z=2400
Z=2000+0.8 ×500=400

52. Merits of Mode :
• Mode is readily comprehensible
and easily calculated
• It is the best representative of data
• It is not at all affected by extreme
value.
• The value of mode can also be
determined graphically.
• It is usually an actual value of an
important part of the series.

53. Demerits of Mode
• It is not based on all
observations.
• It is not capable of further
mathematical manipulation.
• Mode is affected to a great
extent by sampling
fluctuations.
• Choice of grouping has great
influence on the value of mode.

54. Conclusion:
A measure of central tendency is a
measure that tells us where the
middle of a bunch of data lies.
Mean is the most common measure
of central tendency. It is simply the
sum of the numbers divided by the
number of numbers in a set of data.
This is also known as average.

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Median is the number present in
the middle when the numbers in a
set of data are arranged in ascending
or descending order. If the number
of numbers in a data set is even,
then the median is the mean of the
two middle numbers.
Mode is the value that occurs most
frequently in a set of data.