1. Measures of Central Tendency Measures of Variability
The University of the West Indies School of Education Introduction to Statistics
5/12/2019
Dr. Madgerie Jameson
UWI School of Education
Madgerie Jameson
Lessons 3 and 4
Measures of Central Tendency
and
Measures of Variability

2. Topics Measures of central tendency Measures of position
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Measures of central tendency
Mean
Median
Mode
Measures of position
Percentiles
Quartiles
Measures of Variability
Range
Variance
Standard Deviation
Standard Score
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3. OBJECTIVES By the end of the session you will be able to
Compute the mean, median and mode in a given set of scores.
Compute the inter-percentile range of a given set of scores.
Explain the importance of variability in a data set.
Compute the range, standard deviation and variance of a data set.
Compare and Contrast standard deviation and variance.
Compute and interpret the Z score
Describe the normal curve

4. Summary Measures Describing Data Numerically Variation Shape Skewness
Central Tendency
Quartiles
Variation
Shape
Arithmetic Mean
Range
Skewness
Median
Interquartile Range
Mode
Variance
Standard Deviation
Coefficient of Variation

5. Measures of Central Tendency
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Measures of Central Tendency
Mean
Mode
Median
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6. Measures of Central Tendency
Overview
Central Tendency
Arithmetic Mean
Median
Mode
Midpoint of ranked values
Most frequently observed value

7. Mean
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The most common type of average. It is simply the sum of all the values in the group, divided by the number of values.
Population Mean:
Sample Mean:
Frequency Mean:
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8. Computing the Mean
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Compute the average number of shoppers in three different Pennywise locations.
Location
Number of monthly customers
Trincity
2150
Tunapuna
1534
Port of Spain
3564
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9. 5/12/2019 =
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10. Note
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The mean is sometimes represented by the letter M and is also called the typical average, or most central score.
In the formula, a lower case n represents the sample size. An uppercase N represents the population size.
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11. Computing the mean from a grouped frequency distribution
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List all the values in the sample for which the mean is being computed.
List the frequency with which each value occurs
Multiply the value by the frequency
Sum all the values
Divide by the total frequency
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12. Example
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Here is a table that shows the values and frequency in a General Paper test for 100 Form 3 students
Value
Frequency
Value x frequency 97 4
388 11
1034 92 12
1104 91 21
1911 90 30
2700 89
1068 78 9
702 60 1
Total
100
8967
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13. 5/12/2019
The weighted mean is
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14. Median Defined as the midpoint in a set of scores.
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Defined as the midpoint in a set of scores.
No standard formula for computation. The following steps are useful
List all values in order of magnitude. From the highest to lowest or lowest to highest.
Find the middle score
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15. Example Here are the incomes from five different households
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Here are the incomes from five different households
$ , $25 500, $32 456, $ and $37 668
Here is the list ordered from the highest to the lowest.
$ , $54 365, $37,668, $ and $
There are five values the mid-most value is $ and that is the median
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16. When there is an even number the median is
If there are even values, for example, if I added a sixth value $34, 500 to the list.
$ , $54 365, $37,668,$34 500, $32 456
and $
When there is an even number the median is
simply the mean between the two middle
numbers. $37,668 and $ The mean is $ That is the median for the set of six numbers.
If the two middle most values are the same then the median is both numbers, e.g. 9, 7, 6, 6, 5, 4. The median is 6.
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17. Percentile Points
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If you know about medians, you should know about percentile points. They are used to define the percentage of cases equal to and below a certain point. The kth percentile is the number which has k% of the value below it. For example, If a particular score is at the 75th percentile. It means that the score is at or above 75% of the other scores in the distribution. The median is known as the 50th percentile because it is the point below which 50% of the cases in the distribution fall.
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18. Quartiles
Quartiles split the ranked data into 4 segments with an equal number of values per segment
25%
25%
25%
25% Q1 Q2 Q3
The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger
Q2 is the same as the median (50% are smaller, 50% are larger)
Only 25% of the observations are greater than the third quartile

19. Quartile Formulas
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Find a quartile by determining the value in the appropriate position in the ranked data, where
First quartile position: Q1 = (n+1)/4
Second quartile position: Q2 = (n+1)/2 (the median position)
Third quartile position: Q3 = 3(n+1)/4
where n is the number of observed values

20. Quartiles Example: Find the first quartile
Sample Data in Ordered Array:
(n = 9)
Q1 = is in the (9+1)/4 = 2.5 position of the ranked data
so use the value half way between the 2nd and 3rd values,
so Q1 = 12.5
Q1 and Q3 are measures of noncentral location
Q2 = median, a measure of central tendency

21. Quartiles The quartile divide the data into 4 equal regions.
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The quartile divide the data into 4 equal regions.
Q1 : The first quartile is the 25th percentile
Q2 : The second quartile is the 50th percentile or the median
Q3 : the third quartile is the 75th percentile
Q4 : the fourth quartile is the 100th percentile.
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22. Interquartile Range (IQR)
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The distance between the 75th percentile and the 25th percentile. It is the range of the middle 50% of the data set. It is not affected by the extreme outliers or extreme values. To compute the value in the 25th percentile .25* ( n +1) n is the number of values For the 75th percentile .75* ( n +1).
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23. Example Compute the IRQ for the following data
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Compute the IRQ for the following data
18, 33, 58, 67, 73, 93, and 147
The 25th and 75th percentiles are
.25* ( 7 +1) and .75 *( 7 +1).
= 2nd and 6th observations respectively.
IQR = 93 – 33 = 60.
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24. The Mode The value that occurs most frequently in a data set.
5/12/2019
The value that occurs most frequently in a data set.
To compute the mode
List all values in a distribution, but list it only once
Tally the number of times that each value occurs.
The value that occurs the most often is the mode.
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25. Mode A measure of central tendency Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may may be no mode
There may be several modes
No Mode
Mode = 9

26. Example
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An example of the marital status of 300 students resulted in the following distribution of scores The mode is the value that occurs most frequently, which in the above example is single students.
Marital status
Frequency
Single
140
Married
100
Divorce 60
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27. Bimodal
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If every value in a distribution contains the same number of occurrences then there is no mode.
If more than one value appears with equal frequency, the distribution is multimodal.
A data set can be bimodal with two modes.
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28. Example
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Hairstyle
Frequency
Extensions 45
Chemically treated
Natural 12
Bald 7
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In the above example the distribution is bimodal because the frequency values of extensions and chemically treated hair occurs equally.

29. Review Example Five houses on a hill by the beach
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Five houses on a hill by the beach
House Prices: $2,000, , , , ,000

30. Review Example: Summary Statistics
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
House Prices: $2,000,000
500, , , ,000
Sum 3,000,000
Mean: ($3,000,000/5)
= $600,000
Median: middle value of ranked data = $300,000
Mode: most frequent value = $100,000

31. Which measure of location is the “best”?
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Mean is generally used, unless extreme values (outliers) exist
Then median is often used, since the median is not sensitive to extreme values.
Example: Median home prices may be reported for a region – less sensitive to outliers

32. Practice time
5/12/2019
Compute the mean, mode and median for the following data set.
Identify the score in Q1 and Q3.
Calculate the IQR for the data Set.
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33. Reading Scores
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34. Lunch Time
5/12/2019
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35. Coefficient of Variation
Measures of Variation
Variation
Range
Interquartile
Range
Variance
Standard Deviation
Coefficient of Variation
Measures of variation give information on the spread or variability of the data values.
Same center,
different variation

36. Measures of variability
Illustrate the space between scores in a data set. Example: Student A: 10, 12, 15, 18 and 20. Student B: 8, 2, 8, 15, 22 and 28 Same mean of 15 Student B’s scores are more varied than student A’s.

37. Range = Xlargest – Xsmallest
Simplest measure of variation
Difference between the largest and the smallest observations:
Range = Xlargest – Xsmallest
Example:
Range = = 13

38. Disadvantages of the Range
Ignores the way in which data are distributed
Sensitive to outliers
Range = = 5
Range = = 5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
Range = = 4
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = = 119

39. Range
A simple measure of variability. Recap: r = h – l. Range for Student A = 20 – 10 = 10 Range for Student B = 28 – 2 = 26 Disadvantage: not reliable when using large samples.

40. variance The average of the squared deviation from the mean.
Formula for computing variance
Population Variance
2 =
Sample Variance
S2 =

41. Formula Applied Step 1 Compute the mean of the sample
Step 2 Subtract the mean from each score
Step 3 Square each deviation from the mean
Step 4 Add up the squared Deviations from the mean. This number is called the Sum of Squares (ss)

42. Formula applied contd
Step 5 Compute the sample variance by dividing the sum of squares by (n-1)
s2 =
Practice page 5 learning module.

43. Formula for computing Standard Deviation
Describes variability in terms of average distance from the mean.
Formula for computing Standard Deviation
Population SD
Sample SD

44. Calculation Example: Sample Standard Deviation
Sample Data (Xi) :
n = Mean = X = 16
A measure of the “average” scatter around the mean

45. Measuring variation
Small standard deviation
Large standard deviation

46. Comparing Standard Deviations
Data A
Mean = 15.5
S = 3.338
Data B
Mean = 15.5
S = 0.926
Data C
Mean = 15.5
S = 4.570

47. Advantages of Variance and Standard Deviation
Each value in the data set is used in the calculation
Values far from the mean are given extra weight (because deviations from the mean are squared)

48. The normal curve
Shows a distribution of values where the mean and mode are equal to each other.
Characteristics of the Normal Curve
Symmetric
Extends to +/- infinity
Area under the curve is = 1
Can be completely specified by two parameters mean and standard deviation
Empirical Rule: A handy quick estimate of the spread of the data given the mean and standard deviation of the data set.

49. The Empirical Rule
If the data distribution is bell-shaped, then the interval:
contains about 68% of the values in the population or the sample
Data within 1 standard deviation from the mean. The empirical rule state that 68% of data elements are within one standard deviation from the mean.
68%

50. The Empirical Rule contains about 95% of the values in
the population or the sample
contains about 99.7% of the values in the population or the sample
95%
99.7%

51. Skewness
The quality of the distribution that defines the lack of symmetry or lopsidedness of a distribution of scores. The tail of the distribution is longer at one end
Positively skewed: when the right tail of the distribution is longer than the left.
Negatively skewed when the left tail of the distribution is longer than the right.

53. Kurtosis Kurtosis is how flat or peaked a distribution appears
Platykurtic refers to a distribution that is relatively flat compared to the bell curve. More dispers
Leptokurtic refers to a distribution that is relatively peaked compared to the normal or bell curve.
Mesokurtic refers to the bell curve.

55. Standard score or z score
The z score standardises any distribution so that he mean is equal to zero and the standard deviation is equal to one.
Formula to calculate the z score
Z =

56. summary
Measures of variability enhance the ways we describe data. They describe the spread of scores around the average.
The normal curve shows the distribution of values where the mean, median and mode scores are equal to each other.
The z score standardises any distribution so that the mean is equal to zero and the standard deviation is equal to one.

57. References
5/12/2019
Berenson (2004) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Best, J.W. & Kahn, J.V. ( 2006). Research in Education. Boston: Pearson
Phillips, JL (2000). How to think about statistics. New York
Salkind, N. J. (2008) Statistics for people who (think they) hate Statistics. CA: Sage
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