Measures of Central Tendency in Data Analysis

 
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Summary Measures
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Measures of Central Tendency
 
A statistical measure that identifies a single
score as representative for an entire distribution.
The goal of central tendency is to find the single
score that is most typical or most representative
of the entire group
There are three common measures of central
tendency:
the mean
the median
the mode
 
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Calculating the Mean
 
Calculate the mean of the following data:
1   5   4   3   2
Sum the scores (
X)
:
1 + 5 + 4 + 3 + 2 = 15
Divide the sum (
X
 = 15) by the number of scores (N =
5): 15 / 5 = 3
 
Mean = X = 3
 
5
 
Mean (Arithmetic Mean)
 
The most common measure of central tendency
Affected by extreme values (outliers)
 
(continue
d)
 
0   1   2   3   4   5   6   7   8   9   10
 
0   1   2   3   4   5   6   7   8   9   10   12   14
 
Mean = 5
Mean = 6
 
6
 
The Median
 
The 
median
 is simply another name for the 50
th
percentile
It is the score in the middle; half of the scores are larger
than the median and half of the scores are smaller than
the median
 
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How To Calculate the Median
 
Conceptually, it is easy to calculate the median
Sort the data from highest to lowest
Find the score in the middle
middle = (N + 1) / 2
If N, the number of scores is even, the median is the
average of the middle two scores
 
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Median Example
 
What is the median of the following scores:
10   8   14   15   7   3   3   8   12   10   9
Sort the scores:
15   14   12   10   10   9   8   8   7   3   3
Determine the middle score:
middle = (N + 1) / 2 = (11 + 1) / 2 = 6
Middle score = median = 9
 
9
 
Median Example
 
What is the median of the following scores:
24  18  19  42  16  12
Sort the scores:
42  24  19  18  16  12
Determine the middle score:
middle = (N + 1) / 2 = (6 + 1) / 2 = 3.5
Median = average of 3
rd
 and 4
th
 scores:
(19 + 18) / 2 = 18.5
 
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Median
 
 
Not affected by extreme values
 
In an ordered array, the median is the “middle”
number
If n or N is odd, the median is the middle number
If n or N is even, the median is the average of the two
middle numbers
 
0   1   2   3   4   5   6   7   8   9   10
 
0   1   2   3   4   5   6   7   8   9   10   12   14
Median = 5
Median = 5
 
Measures of Central Tendency
 
Mean 
… the most frequently used but is sensitive
to extreme scores
e.g. 1  2  3  4  5  6  7  8  9  10
Mean = 5.5 (median = 5.5)
e.g. 1  2  3  4  5  6  7  8  9  20
Mean = 6.5 (median = 5.5)
e.g. 1  2  3  4  5  6  7  8  9  100
Mean = 14.5 (median = 5.5)
 
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Mode
 
 
Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may be no mode
There may be several modes
 
0   1   2   3   4   5   6   7   8   9   10   11   12   13   14
Mode = 9
 
0   1   2   3   4   5   6
No Mode
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The Shape of Distributions
Distributions can be either 
symmetrical
 or
skewed
, depending on whether there are more
frequencies at one end of the distribution than
the other.
 
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Symmetrical
Distributions
 
A distribution is symmetrical if the frequencies at
the right and left tails of  the distribution are
identical, so that if it is divided into two halves,
each will be the mirror image of the other.
 
 In a  symmetrical distribution the mean, median,
and mode are identical.
 
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Distributions
 
Bell-Shaped (also
known as symmetric”
or “normal”)
 
Skewed:
positively (skewed to
the right) – it tails off
toward larger values
negatively (skewed to
the left) – it tails off
toward smaller values
 
16
 
Skewed Distribution
F
ew extreme values on one side of the distribution or on the
other.
 
Positively skewed
 
distributions:  distributions
which have few extremely high values
(Mean>Median)
Negatively skewed distributions:  distributions
which have  few extremely low
values(Mean<Median)
17
Mean=1.13
Median=1.0
 
18
 
Mean=3.3
Median=4.0
 
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Choosing  a Measure of Central Tendency
 
IF variable is Nominal..
Mode
IF variable is Ordinal...
Mode or Median(or both)
IF variable is Interval-Ratio and distribution is
Symmetrical…
Mode, Median or Mean
IF variable is Interval-Ratio and distribution is
Skewed…
Mode or Median
 
EXAMPLE:
 
 
   
(1) 7,8,9,10,11   n=5,    x=45,        =45/5=9
 
   (2) 3,4,9,12,15   n=5,   x=45,         =45/5=9
 
   (3) 1,5,9,13,17   n=5,   x=45,         =45/5=9
 
   S.D. :  (1) 1.58 (2) 4.74 (3) 6.32
 
 
 
 
 
 
 
 
Measures of Dispersion
Or
Measures of variability
 
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Locating Percentiles in a Frequency
Distribution
A percentile is a score below which a specific percentage of the distribution
falls(the median is the 50th percentile.
 
The 75th percentile  is a score below which 75% of the cases fall.
 
The median is the 50th percentile: 50% of the cases fall below it
 
Another type of percentile :The quartile  lower quartile is 25th percentile and
the upper quartile is the 75th percentile
 
 
50th
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80th
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Quantifying Uncertainty
Standard deviation
:  measures the variation of a variable in the
sample.
Technically,
Example:
Data: X = {6, 10, 5, 4, 9, 8};         N = 6
 
Total: 42
 
Total: 28
 
 Standard Deviation:
 
Mean:
 
Variance:
 
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Measures of central tendency, including mean and median, play vital roles in summarizing and interpreting data. The mean is the average calculated by summing all values and dividing by the count, while the median is the middle score when data is arranged in order. These measures provide insight into the typical value and help in making informed decisions based on data distributions.


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  1. INVESTIGATION Data Collection Inferential Statistics Data Presentation Descriptive Statistics Inferential statistics Estimation Hypothesis Measures of Location Measures of Dispersion Measures of Skewness & Kurtosis Tabulation Diagrams Graphs Testing Univariate analysis Point estimate Interval estimate Multivariate analysis

  2. Summary Measures Describing Data Numerically Central Tendency Quartiles Variation Shape Arithmetic Mean Range Skewness Median Interquartile Range Mode Variance Geometric Mean Standard Deviation Harmonic Mean 2

  3. Measures of Central Tendency A statistical measure that identifies a single score as representative for an entire distribution. The goal of central tendency is to find the single score that is most typical or most representative of the entire group There are three common measures of central tendency: the mean the median the mode 3

  4. Calculating the Mean Calculate the mean of the following data: 1 5 4 3 2 Sum the scores ( X): 1 + 5 + 4 + 3 + 2 = 15 Divide the sum ( X = 15) by the number of scores (N = 5): 15 / 5 = 3 Mean = X = 3 4

  5. Mean (Arithmetic Mean) (continue d) The most common measure of central tendency Affected by extreme values (outliers) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5 Mean = 6 5

  6. The Median The median is simply another name for the 50th percentile It is the score in the middle; half of the scores are larger than the median and half of the scores are smaller than the median 6

  7. How To Calculate the Median Conceptually, it is easy to calculate the median Sort the data from highest to lowest Find the score in the middle middle = (N + 1) / 2 If N, the number of scores is even, the median is the average of the middle two scores 7

  8. Median Example What is the median of the following scores: 10 8 14 15 7 3 3 8 12 10 9 Sort the scores: 15 14 12 10 10 9 8 8 7 3 3 Determine the middle score: middle = (N + 1) / 2 = (11 + 1) / 2 = 6 Middle score = median = 9 8

  9. Median Example What is the median of the following scores: 24 18 19 42 16 12 Sort the scores: 42 24 19 18 16 12 Determine the middle score: middle = (N + 1) / 2 = (6 + 1) / 2 = 3.5 Median = average of 3rd and 4th scores: (19 + 18) / 2 = 18.5 9

  10. Median Not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 Median = 5 In an ordered array, the median is the middle number If n or N is odd, the median is the middle number If n or N is even, the median is the average of the two middle numbers 10

  11. Measures of Central Tendency Mean the most frequently used but is sensitive to extreme scores e.g. 1 2 3 4 5 6 7 8 9 10 Mean = 5.5 (median = 5.5) e.g. 1 2 3 4 5 6 7 8 9 20 Mean = 6.5 (median = 5.5) e.g. 1 2 3 4 5 6 7 8 9 100 Mean = 14.5 (median = 5.5)

  12. Mode Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may be no mode There may be several modes 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 No Mode Mode = 9 12

  13. The Shape of Distributions Distributions can be either symmetrical or skewed, depending on whether there are more frequencies at one end of the distribution than the other. ? 13

  14. Symmetrical Distributions A distribution is symmetrical if the frequencies at the right and left tails of the distribution are identical, so that if it is divided into two halves, each will be the mirror image of the other. In a symmetrical distribution the mean, median, and mode are identical. 14

  15. Distributions Bell-Shaped (also known as symmetric or normal ) skew6 Skewed: positively (skewed to the right) it tails off toward larger values negatively (skewed to the left) it tails off toward smaller values 15

  16. Skewed Distribution Few extreme values on one side of the distribution or on the other. Positively skeweddistributions: distributions which have few extremely high values (Mean>Median) Negatively skewed distributions: distributions which have few extremely low values(Mean<Median) 16

  17. Positively Skewed Distribution GOVT INVESTIGATE WORKERS ILLEGAL DRUG USE Mean=1.13 500 400 Median=1.0 300 200 Frequency 100 Std. Dev = .39 Mean = 1.1 N = 474.00 0 1.0 2.0 3.0 4.0 GOVT INVESTIGATE WORKERS ILLEGAL DRUG USE 17

  18. Negatively Skewed distribution FAVOR PREFERENCE IN HIRING BLACKS 600 Mean=3.3 500 400 Median=4.0 300 200 Frequency 100 Std. Dev = .98 Mean = 3.3 N = 908.00 0 1.0 2.0 3.0 4.0 FAVOR PREFERENCE IN HIRING BLACKS 18

  19. Choosing a Measure of Central Tendency IF variable is Nominal.. Mode IF variable is Ordinal... Mode or Median(or both) IF variable is Interval-Ratio and distribution is Symmetrical Mode, Median or Mean IF variable is Interval-Ratio and distribution is Skewed Mode or Median 19

  20. EXAMPLE: x (1) 7,8,9,10,11 n=5, x=45, =45/5=9 (2) 3,4,9,12,15 n=5, x=45, =45/5=9 x x (3) 1,5,9,13,17 n=5, x=45, =45/5=9 S.D. : (1) 1.58 (2) 4.74 (3) 6.32

  21. Measures of Dispersion Or Measures of variability 21

  22. Series I: 70 70 70 70 70 70 70 70 70 70 Series II: 66 67 68 69 70 70 71 72 73 74 Series III: 1 19 50 60 70 80 90 100 110 120 22

  23. Measures of Variability A single summary figure that describes the spread of observations within a distribution. 23

  24. Measures of Variability Range Difference between the smallest and largest observations. Interquartile Range Range of the middle half of scores. Variance Mean of all squared deviations from the mean. Standard Deviation Rough measure of the average amount by which observations deviate from the mean. The square root of the variance. 24

  25. Variability Example: Range Marks of students 52, 76, 100, 36, 86, 96, 20, 15, 57, 64, 64, 80, 82, 83, 30, 31, 31, 31, 32, 37, 38, 38, 40, 40, 41, 42, 47, 48, 63, 63, 72, 79, 70, 71, 89 Range: 100-15 = 85 25

  26. Quartiles Q1, Q2, Q3 divides ranked scores into four equal parts 25% 25% 25% 25% Q3 Q2 (median) Q1 (minimum) (maximum) 26

  27. Quartiles: n+1 th 4 2(n+1) = 4 3(n+1) th 4 Q Q 1 = n+1 2 2 = th Q 3 = Inter quartile : IQR = Q3 Q1 27

  28. Inter quartile Range The inter quartile range is Q3-Q1 50% of the observations in the distribution are in the inter quartile range. The following figure shows the interaction between the quartiles, the median and the inter quartile range. 28

  29. Inter quartile Range 29

  30. Percentiles and Quartiles Maximum is 100th percentile: 100% of values lie at or below the maximum Median is 50th percentile: 50% of values lie at or below the median Any percentile can be calculated. But the most common are 25th (1st Quartile) and 75th (3rd Quartile) 30

  31. 31

  32. Locating Percentiles in a Frequency Distribution A percentile is a score below which a specific percentage of the distribution falls(the median is the 50th percentile. The 75th percentile is a score below which 75% of the cases fall. The median is the 50th percentile: 50% of the cases fall below it Another type of percentile :The quartile lower quartile is 25th percentile and the upper quartile is the 75th percentile 32

  33. Locating Percentiles in a Frequency Distribution 25% included N U M B E R O F r e q u e n c y P e r c e n t Va lid P e r c e n t C u m P e r c e n t F C H IL D R E N 0 2 3 4 5 6 7 E I G HT O T o ta l N A T o ta l Va lid u la tive here 25th percentile 50% included 50th percentile here 80th 80% included percentile here R MO R E 33

  34. VARIANCE: Deviations of each observation from the mean, then averaging the sum of squares of these deviations. STANDARD DEVIATION: ROOT- MEANS-SQUARE-DEVIATIONS 34

  35. Standard Deviation To undo the squaring of difference scores, take the square root of the variance. Return to original units rather than squared units. 35

  36. Quantifying Uncertainty Standard deviation: measures the variation of a variable in the sample. Technically, N = = 2 ( ) s x x 1 i 1 N 1 i 36

  37. Example: Data: X = {6, 10, 5, 4, 9, 8}; N = 6 Mean: X 2) X X 6 10 5 4 9 8 ( X X = X 42= = 7 X -1 3 -2 -3 2 1 1 9 4 9 4 1 6 N Variance: 2 ( ) X X 28 6 = = = 2 4.67 s N Standard Deviation: = s = = 2 . 4 67 . 2 16 s Total: 42 Total: 28

  38. Calculation of Variance & Standard deviation Using the deviation & computational method to calculate the variance and standard deviation Example: 3,4,4,4,6,7,7,8,8,9 ; Given n=10; Sum= 60; Mean = 6 = 2 ( ) X X S n ) 6 4 ( + ) 6 4 ( + ) 6 4 ( + ) 6 6 ( + ) 6 7 ( + ) 6 7 ( + ) 6 + ) 6 + ) 6 + ) 6 2 2 2 2 2 2 2 2 2 2 3 ( 8 ( 8 ( 9 ( = S 10 40 = = = ; 0 . 2 var 4 S iance 10 X2 X 2 2 ( ) n X X = 3 4 4 4 6 7 7 8 8 9 9 S 2 n 16 16 16 36 49 49 64 64 81 2 10 ( 400 ) ( 60 ) = S 2 10 4000 3600 = S 100 = 0 . 4 S = = , 0 . 2 var 4 S iance Sum: 60 Sum: 400 38

  39. WHICH MEASURE TO USE ? DISTRIBUTION OF DATA IS SYMMETRIC ---- USE MEAN & S.D., DISTRIBUTION OF DATA IS SKEWED ---- USE MEDIAN & QUARTILES 39

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