Measures of Central Tendency

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Measures of Central Tendency

• By Rahul Jain

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The Motivation

• Measure of central tendency are used to describe
  the typical member of a population.
• Depending on the type of data, typical could have
  a variety of best meanings.
• We will discuss four of these possible choices.

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4 Measures of Central Tendency

• Mean the arithmetic average. This is used for
  continuous data.
• Median a value that splits the data into two
  halves, that is, one half of the data is smaller
  than that number, the other half larger. May be
  used for continuous or ordinal data.
• Mode this is the category that has the most
  data. As the description implies it is used for
  categorical data.
• Midrange not used as often as the other three,
  it is found by taking the average of the lowest
  and highest number in the data set. Also
  primarily used for continuous data.

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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 of the "middle"
  or "expected" value of the data set.

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Mean

• To find the mean, add all of the values, then
  divide by the number of values.
• The lower case, Greek letter mu is used for
  population mean.
• An x with a bar over it, read x-bar, is used
  for sample mean.

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Mean Example
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Arithmetic Mean of Group Data

• if are the mid-values
  and
• are the corresponding
  frequencies, where the subscript k stands for
  the number of classes, then the mean is

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Exercise-1 Find the Arithmetic Mean
Class Frequency (f) x fx
20-29 3 24.5 73.5
30-39 5 34.5 172.5
40-49 20 44.5 890
50-59 10 54.5 545
60-69 5 64.5 322.5
Sum N43 2003.5
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Median

• The median is a number chosen so that half of the
  values in the data set are smaller than that
  number, and the other half are larger.
• To find the median
• List the numbers in ascending order
• If there is a number in the middle (odd number of
  values) that is the median
• If there is not a middle number (even number of
  values) take the two in the middle, their average
  is the median

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Median Example
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Median

• The implication of this definition is that a
  median is the middle value of the observations
  such that the number of observations above it is
  equal to the number of observations below it.

If n is Even
If n is odd
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Median of Group Data

• 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-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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Mode

• The mode is simply the category or value which
  occurs the most in a data set.
• If a category has radically more than the others,
  it is a mode.
• Generally speaking we do not consider more than
  two modes in a data set.
• No clear guideline exists for deciding how many
  more entries a category must have than the others
  to constitute a mode.

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Obvious Example

• There is obviously more yellow than red or blue.
• Yellow is the mode.
• The mode is the class, not the frequency.

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Bimodal
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No Mode

• Category Frequency
• 1 51
• 2 51
• 3 66
• 4 62
• 5 65
• 6 57
• 7 47
• 8 43
• 64
• Although the third category is the largest, it is
  not sufficiently different to be called the mode.

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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
Median 4
Mode 4
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Mode of Group Data

• L1 Lower boundary of modal class
• ?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
• h Interval 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
5-9 7 12 84
10-14 12 7 84
15-19 17 5 85
20-24 22 0 0
Total Total n 30 ?(fx) 265
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Midrange

• The midrange is the average of the lowest and
  highest value in the data set.
• This measure is not often used since it is based
  strictly on the two extreme values in the data.

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Midrange Example
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Measures of Variation
Same mean, but y varies more than x.
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Three Measures of Variation

• While there are other measures, we will look at
  only three
• Variance
• Standard deviation
• Coefficient of variation
• Population mean and sample mean use an identical
  formula for calculation.
• There is a minor difference in the formulas for
  variation.

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Population Variance

• The population variance, s2, is found using
  either of the formulas to the right.
• The differences are squared to prevent the sum
  from being zero for all cases.
• N is the size of the population, µ is the
  population mean.
• Note that variance is always positive if x can
  take on more than one value.

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Population Standard Deviation

• The standard deviation can be thought of as the
  average amount we could expect the xs in the
  population to differ from the mean value of the
  population.
• To get the standard deviation, simply take the
  square root of the variance.

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Sample Variance

• The sample variance, s2, is found using either of
  the formulas to the right.
• The differences are squared to prevent the sum
  from being zero for all cases.
• The sample size is n, x-bar is the sample mean.
• Note that n-1 is used rather than n. This
  adjustment prevents bias in the estimate.

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Sample Standard Deviation

• Just like the standard deviation of a population,
  to find the standard deviation of a sample, take
  the square root of the sample variance.

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Coefficient of Variation

• The measures discussed so far are primarily
  useful when comparing members from the same
  population, or comparing similar populations.
• When looking at two or more dissimilar
  populations, it doesnt make any more sense to
  compare standard deviations than it does to
  compare means.

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Coefficient of Variation Cont.

• Example 1 Weight loss programs A and B.
• Two different programs with the same goal and
  target population.
• While program B averages more weight loss, it
  also has less consistent results.

A B
Mean (weight loss per month) 20 25
Standard deviation 15 30
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Coefficient of Variation Cont.

• Example 2 Weight loss program A and tax refund
  B.
• Two different programs with different goals and
  different target populations.
• We know that average weight loss and average tax
  refund are not comparable. Are the standard
  deviations comparable?

A B
Mean 20 650
Standard deviation 15 30
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Coefficient of Variation Cont.

• In the last example we can see an argument that
  standard deviation does not give the complete
  picture.
• The coefficient of variation addresses this issue
  by establishing a ratio of the standard deviation
  to the mean. This ratio is expressed as a
  percentage.

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Coefficient of Variation Cont.

• Looking at the two examples. We see that in both
  cases the standard deviation for B is twice that
  of A.
• In the first example we have almost twice the
  relative variation in B.
• In the second example, we have a little over 16
  times as much variation in A.

A B
CV Example 1 75 120
CV Example 2 75 4.6
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Measures of Position
The dot on the left is at about -1, the dot on
the right is at approximately 0.8. But where are
they relative to the rest of the values in this
distribution.
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Quartiles, Percentiles and Other Fractiles

• We will only consider the quartile, but the same
  concept is often extended to percentages or other
  fractions.
• The median is a good starting point for finding
  the quartiles.
• Recall that to find the median, we wanted to
  locate a point so that half of the data was
  smaller, and the other half larger than that
  point.

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Quartile

• For quartiles, we want to divide our data into 4
  equal pieces.

Suppose we had the following data set (already in
order) 2 3 7 8 8 8 9 13 17 20 21 21
Choosing the numbers 7.5, 8.5, and 18.5 as
markers would Divide the data into 4 groups, each
with three elements. These numbers would be the
three quartiles for this data set.
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Quartiles Continued

• Conceptually, this is easy, simply find the
  median, then treat the left hand side as if it
  were a data set, and find its median then do the
  same to the right hand side.
• This is not always simple. Consider the following
  data set.
• 3 3 3 3 3 5 6 8 8 8 8 8 9
• The first difficulty is that the data set does
  not divide nicely.
• Using the rules for finding a median, we would
  get quartiles of 3, 6 and 8.
• The second difficulty is how many of the 3s are
  in the first quartile, and how many in the second?

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Quartiles Continued

• For this course, lets pretend that this is not
  an issue.
• I will give you the quartiles.
• I will not ask how many are in a quartile.

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Interquartile Range

• One method for identifying these outliers,
  involves the use of quartiles.
• The interquartile range (IQR) is Q3 Q1.
• All numbers less than Q1 1.5(IQR) are probably
  too small.
• All numbers greater than Q3 1.5(IQR) are
  probably too large.

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Measures of Variation Variance Standard
Deviationfor GROUPED DATA

• The grouped variance is
• The grouped standard deviation is

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Example 3-24 (p130) Miles Run per Week

• Find the variance and the standard deviation for
  the frequency distribution below. The data
  represents the number of miles that 20 runners
  ran during one week.

Class f Xm fXm f(Xm X)
5.5 10.5 10.5 15.5 15.5 20.5 20.5 25.5 25.5 30.5 30.5 35.5 35.5 40.5 1 2 3 5 4 3 2 20
8 13 18 23 28 33 38
18 8 213 26 318 54 523 115 428
108 333 99 238 76 SfXm 486
1(8-24.3)2 265.69 2(13-24.3)2
255.38 3(18-24.3)2 119.07 5(23-24.3)2
8.45 4(28-24.3)2 54.76 3(33-24.3)2
227.07 2(38-24.3)2 375.38 S f(Xm X) 1305.80
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Mean Deviation

• The mean deviation is an average of absolute
  deviations of individual observations from the
  central value of a series. Average deviation
  about mean
• k Number of classes
• xi Mid point of the i-th class
• fi frequency of the i-th class

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Coefficient of Mean Deviation

• The third relative measure is the coefficient of
  mean deviation. As the mean deviation can be
  computed from mean, median, mode, or from any
  arbitrary value, a general formula for computing
  coefficient of mean deviation may be put as
  follows

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Coefficient of Range

• The coefficient of range is a relative measure
  corresponding to range and is obtained by the
  following formula
• where, L and S are respectively the largest
  and the smallest observations in the data set.

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Coefficient of Quartile Deviation

• The coefficient of quartile deviation is computed
  from the first and the third quartiles using the
  following formula

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Assignment-1

• Find the following measurement of dispersion from
  the data set given in the next page
• Range, Percentile range, Quartile Range
• Quartile deviation, Mean deviation, Standard
  deviation
• Coefficient of variation, Coefficient of mean
  deviation, Coefficient of range, Coefficient of
  quartile deviation

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Data for Assignment-1
Marks No. of students Cumulative frequencies
40-50 6 6
50-60 11 17
60-70 19 36
70-80 17 53
80-90 13 66
90-100 4 70
Total 70