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8/3/2019 Chapter2 Measure of Central Tendency
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Measures of Central Tendency
Measures of Location
Descriptive Statistics
Measures of Symmetry
Measures of Peakdness
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Measures of Central Tendency
The central tendency is measured by averages.
These describe the point about which the
various observed values cluster.
In mathematics, an average, or central
tendency of a data set refers to a measure ofthe "middle" or "expected" value of the data
set.
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Measures of Central Tendency
Arithmetic Mean
Geometric Mean
Harmonic Mean
Median
Mode
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Choosing a measure of central tendency the level of measurement of the variable
concerned (nominal, ordinal, interval or ratio);
the shape of the frequency distribution;
what is to be done with the figure obtained.
Measure of central tendency
The mean is really suitable only for ratio andinterval data. For ordinal variables, where thedata can be ranked but one cannot validly talk
of `equal differences' between values, themedian, which is based on ranking, may beused. Where it is not even possible to rank thedata, as in the case of a nominal variable, themode may be the only measure available.
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Summary
1. The purpose of central tendency is to determine the single valuethat best represents the entire distribution of scores. The three
standard measures of central tendency are the mode, the median,
and the mean.
2. The mean is the arithmetic average. It is computed by summing all
the scores and then dividing by the number of scores. Conceptually,
the mean is obtained by dividing the total (IX) equally among the
number of individuals (N or n). Although the calculation is the same
for a population or a sample mean, a population mean is identified
by the symbol and a sample mean is identified by X.
3. Changing any score in the distribution will cause the mean to be
changed. When a constant value is added to (or subtracted from)every score in a distribution, the same constant value is added to
(or subtracted from) the mean. If every score is multiplied by a
constant, the mean will be multiplied by the same constant. In
nearly all circumstances, the mean is the best representative value
and is the preferred measure of central tendency.
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Summary1. The median is the value that divides a distribution exactly in half.
The median is the preferred measure of central tendency when
a distribution has a few extreme scores that displace the value
of the mean. The median also is used when there are
undetermined (infinite) scores that make it impossible to
compute a mean.
2. The mode is the most frequently occurring score in a .
frequency distribution graph. For data measured on a nominal
scale, the mode is the appropriate measure of central tendency.
It is possible for a distribution to have more than one mode.
3. For symmetrical distributions, the mean will equal the median.If there is only one mode, then it will have the same value, too.
4. For skewed distributions, the mode will be located toward the
side where the scores pile up, and the mean will be pulled
toward the extreme scores in the tail. The median will be
located between these two values.
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Arithmetic Mean
The arithmetic mean is the sum of a set of
observations, positive, negative or zero,
divided by the number of observations. If we
have n real numbers
,.......,,,, 321 nxxxx
their arithmetic mean, denoted by , can be
expressed as:
n
xxxxx n
.............321
n
x
x
n
i
i 1
x
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Arithmetic Mean of Group Data
if are the mid-values and
are the corresponding
kzzzz .,,.........,, 321
kffff ,........,,, 321
,
the number of classes, then the mean is
i
ii
f
zfz
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Geometric Mean
Geometric mean is defined as the positive root of theproduct of observations. Symbolically,
n/1
It is also often used for a set of numbers whose values aremeant to be multiplied together or are exponential in nature,
such as data on the growth of the human population orinterest rates of a financial investment.
Find geometric mean of rate of growth: 34, 27, 45, 55, 22, 34
n321
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Geometric mean of Group data
If the n non-zero and positive variate-values
occur times, respectively,
then the eometric mean of the set of
nxxx ,........,,
21 nfff ,.......,, 21
observations is defined by:
Nn
i
f
iN
f
n
ffin xxxxG
1
1
1
2121
n
i
ifN1
Where
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Geometric Mean (Revised Eqn.)
321 fff
Ungroup Data Group Data
2 n
n
ii
xLogN
AntiLogG
1
1
n
iiixLogf
NAntiLogG
1
1
321 n
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Harmonic Mean
Harmonic mean (formerly sometimes called thesubcontrary mean) is one of several kinds of
average.
Typically, it is appropriate for situations when theaverage of rates is desired. The harmonic mean isthe number of variables divided by the sum of thereciprocals of the variables. Useful for ratios suchas speed (=distance/time) etc.
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Harmonic Mean Group Data
The harmonic mean H of the positive real
numbers x1,x2, ..., xn is defined to be
n
i i
i
x
f
n
H
1
n
i ix
n
H
1
1
Ungroup Data Group Data
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Exercise-1: Find the Arithmetic ,
Geometric and Harmonic Mean
Class Frequency
(f)
x fx f Log x f / x
20-29 3 24.5 73.5 4.17 8.17
30-39 5 34.5 172.5 7.69 6.9
40-49 20 44.5 890 32.97 2.23
50-59 10 54.5 545 17.37 5.45
60-69 5 64.5 322.5 9.05 12.9
Sum N=43 2003.5 71.24 35.64
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Weighted Mean
The Weighted mean of the positive real numbers x1,x2,
..., xn with their weight w1,w2, ..., wn is defined to be
n
i
i
n
i
ii
w
xw
x
1
1
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Median
The implication of this definition is that amedian is the middle value of the
observations such that the number of o servat ons a ove t s equa to t e num erof observations below it.
)1(2
1
n
e XM
122
2
1nne XXM
If n is oddIf n is Even
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Median of Group Data
F
n
f
hLM
o
oe2
L0 = Lower class boundary of the median
class
h = Width of the median class f0 = Frequency of the median class
F = Cumulative frequency of the pre-
median class
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Steps to find Median of group data
1. Compute the less than type cumulative frequencies.
2. Determine N/2 , one-half of the total number of cases.
3. Locate the median class for which the cumulative frequency is
more than N/2 .
4. Determine the lower limit of the median class. This is L0.
5. Sum the frequencies of all classes prior to the median class.
This is F.
6. Determine the frequency of the median class. This is f0.
7. Determine the class width of the median class. This is h.
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Example-3:Find Median
Age in years Number of births Cumulative number of
births
14.5-19.5 677 677
19.5-24.5 1908 2585
24.5-29.5 1737 4332
29.5-34.5 1040 5362
34.5-39.5 294 5656
39.5-44.5 91 5747
44.5-49.5 16 5763
All ages 5763 -
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Example-2: Find Mean, Median and
Mode of Ungroup Data
The weekly pocket money for 9 first year pupils was
found to be:
3 , 12 , 4 , 6 , 1 , 4 , 2 , 5 , 8
Mean
5
Mode
4
Median
4
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Mode of Group Data
hLM21
110
1
1 = difference of frequency between
modal class and class before it
2 = difference of frequency between
modal class and class after
H = class interval
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Steps of Finding Mode
Find the modal class which has highest frequency
L0 = Lower class boundary of modal class
=
1 = difference of frequency of modal
class and class before modal class
2 = difference of frequency of modal class and
class after modal class
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Example -4: Find Mode
Slope Angle
()
Midpoint (x) Frequency (f) Midpoint x
frequency (fx)
0-4 2 6 12
-
10-14 12 7 84
15-19 17 5 85
20-24 22 0 0
Total n = 30 (fx) = 265
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Measures of Central Tendency
Consider the Measurements and Frequency Table
87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77,
73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75,
84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82,
85, 86, 81, 72, 69, 65, 71, 73, 78, 81, 77, 74, 77, 72, 68
Class Class Midpoint Total Frequency
64.5 - 69.5 67 6 0.100
69.5 74.5 72 11 0. 183
74.5 79.5 77 20 0.333
79.5 84.5 82 13 0.217
84.5 89.5 87 9 0.150
89.5 94.5 92 1 0.0167
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Measures of Central Tendency
For the 60 temperature readings in this population we obtain:
87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77,
73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75,
84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82,85 86 81 72 69 65 71 73 78 81 77 74 77 72 68
= (87+85+ 79 +.+72+68)/60 = 4751/60 = 79.183
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Measures of Central Tendency
A third measure of central tendency is the median
The median of a population of size N is found by
1. Arranging the individual measurements in ascending order, and
2. If N is odd, selecting the value in the middle of this list as the median (there
will be the same number of values above and below the median)
3. If N is even find the values at position N/2 and N/2 + 1 in this list (call them
xN/2 an xN/2+1 an e me an e g ven y e ormu a me an = xN/2 +
xN/2+1)/2 or be the value halfway between these two measurements.
Note! When N is even the median will usually not be an actual value in the
population
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Measures of Central Tendency
We now find the median of the population of temperature readings
87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77,
73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75,
84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82,
85, 86, 81, 72, 69, 65, 71, 73, 78, 81, 77, 74, 77, 72, 68
Arrange these 60 measurements in ascending order
65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73, 74, 74, 75, 75, 75,
75, 76, 76, 77, 77, 77, 77, 77, 77, 78, 78, 78, 78, 79, 79, 79, 79, 80, 81, 81,81, 81, 81, 81, 81, 81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92
Since N/2 = 30 and both the 30th and 31st values in the list are the same, we obtain
median = 78
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Measures of Central Tendency
One further parameter of a population that may give some indication of central
tendency of the data is the mode
Define: mode = most frequently occurring value in the
population
From the previous data we see:
65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73, 74, 74, 75, 75, 75, 75,
76, 76, 77, 77, 77, 77, 77, 77, 78, 78, 78, 78, 79, 79, 79, 79, 80, 81, 81, 81, 81,
81, 81, 81, 81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92
That the value 81 occurs 8 times mode = 81
Note! If two different values were to occur most frequently, the distribution would be
bimodal. A distribution may be multi-modal.
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Measures of Central Tendency
From the table we obtain
Class Class Midpoint (x) Total (f) Frequency f*x
64.5 - 69.5 67 6 0.100 402
69.5 74.5 72 11 0. 183 792
74.5 79.5 77 20 0.333 1540
79.5 84.5 82 13 0.217 1066
84.5 89.5 87 9 0.150 783
89.5 94.5 92 1 0.0167 92
60 4675
= i(fi * xi) / i fi = 4675/60 = 77.917
The small discrepancy between these two values for the mean is due to the
way the data is accumulated into classes. The mean of the raw data is more
accurate, the mean of the tabulated data is often more convenient to obtain.
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Numerical DataProperties & Measures
Numerical Data
Properties
Central
MeanMean
MedianMedian
ModeMode
Tendency
RangeRange
VarianceVariance
Standard DeviationStandard Deviation
ar a on
SkewSkew
ape