Central Tendency Measures and Variability

1. Central Tendency and Variability
Dr. Pedro L. Martinez

4. Course A Course B Course C
Student
Hours
per
week
Student
Hours
per
week
Student
Hours
per
week
Joe 9 Hannah 5 Meena 6
Peter 7 Ben 6 Sonia 6
Zoey 8 Iggy 6 Kim 7
Ana 8 Louis 6 Mike 5
Jose 7 Keesha 7 Jamie 6
Lee 9 Lisa 6 Ilana 6
Joshua 8 Mark 5 Lars 5
Ravi 9 Ahmed 5 Nick 20
Kristen 8 Jenny 6 Liz 5
Loren 1 Erin 6 Kevin 6
Suppose you wanted to know how many hours students spend studying for three distinct courses. The researcher does
a survey of ten students in each of the courses. On the survey, he asks the students to write down the number of hours
per week they spend studying for that course. The data look like this:

5. Response:
Strongly
Disagree
Disagree
Neither Agree
nor Disagree
Agree Strongly Agree
Code: 1 2 3 4 5
The Likert scale is an example of an ordinal scale. Consider five possible responses to a question, Is our
instructor is an excellent teacher?, with answers on this scale. A 5 will guarantee you an passing grade in
this course!

6. If you worked for Pizza Hut, you might want to know what are
the preferences between men and women when choosing
pizza toppings.
Men Preferences
______________
______________
______________
Women Preferences
________________
________________
________________

7. If you want to summarize the above examples you mat be able
to summarize data, by using:
Measures of central tendency measures: They tell you
about the typical scores
or
Measures of variability: They tell you about how
scores are spread out

12. We buy:
100 size 4pairs
100 size 5pairs
100 size 6 pairs
100 size 7 pairs
100 size 8 pairs
100 size 9 pairs
100 size 10 pairs
100 size 11 pairs
100 size 12 pairs
100 size 13 pairs
100 size 14 pairs
4 5 6 7 8 9 10 11 12 13 14
Shoes On Sale

14. 9 1 0 1 1 1 2 1 3 1 46 7 854
Hint – This is the Normal Curve and the
Center is the Mean

15. When we had the same number of all the sizes – WE DID NOT
PAY ATTENTION TO THE FACT that there aren’t equal
numbers of feet at each shoe size.
Conclusion – we need to know two
things:
1. The Typical Score: Central Tendency
2. How are the scores spread out: Variability

17. Frequency
MEN WOMEN

18. Frequency
Mean
4 9
14
Shoe Size
Hints:
•This is the normal curve and the line in the middle is the
mean.
A little box on the graph represents a score.

19. Shoe Size of Short Jockeys Shoe Size of Tall Basketball Players
99
Nuances:
Distributions can come in various other shapes:
1: Skewed with the more scores to the left or right.
Positive Skew Negative Skew

20. 2. Bimodal (also multimodal) with more
than one peak
Shoe Size of Men and Women combined.
Which hump do you think is the male peak?

22. Central Tendency
Mode
The most common score
or the score with the
highest frequency.
Used with
Nominal, Ordinal,
Measurement data
Median
Divides distribution in
half. 50 % of scores
above median & 50%
below median
Used with Ordinal &
Measurement data.
Mean
Arithmetic Average.
Take all the scores, add
them up and divide by
number of scores
Mean = ΣX/N
For Measurement
data

25. Variability
Range
Highest score minus the
Lowest Score.
Used with
Ordinal, Measurement
data
Subject to extremes
Standard Deviation
√[Σ(X-mean)2
/N]
For Measurement data
Note: the Variance
equals the square of the
SD
The standard deviation is the most commonly used
measure of variable with measurement data.

26. Me an
St a n d a r d
D e v i a t i o n

27. Mean
St a n d a rd
D e v i a t i o n
9 11
7 9
68.26% of the shoe sizes between 7 and 11
Let’s look at our shoe data
If you had calculated the standard deviation of the distribution of shoe
sizes and found out it was 2.0* you would know that 68.26 % of males
probably have shoe sizes between 7 and 11. This 9 + 1 SD or 9 + 2

32. Why is the Normal Distribution so Important?
Researchers sometime want to simply
describe the characteristics of a
population not necessarily the scores.
Example: If I wanted to learn about the
eating habits of college students, what
would I do?
How would I conduct my study?

33. Descriptive Statistics
 Descriptive statistics are used to describe the
characteristics of a sample.
 Central tendency statistics tells us more about
the sample. It helps u s to determine how
probable are the findings. This is referred to as
inferential statistics.
 Inferential statistics are used to make
predictions about the population based on a
sample.

34. What is the daily calorie consumption of men vs.
women?
 How would inferential statistics help us to look
at this research?
 Null hypothesis
 Relationship[ between sampling and the
normal distribution
 A) Representative Sampling
 B) Random Sampling
 C) Convenience Sampling

35. Curves
 Skewed Curves (positive, negative)
 Kurtosis-height of the curve
 Peak is higher than the normal distribution
then is said to be leptokurtic
 When is flatter then is said to be platykurtic

36. Kurtosis
If 68%of scores fall within the mean in a normal
curve what happens in the following curves?

37. Normal Curve
6
When the mean, median and mode are equal, you
will have a normal or bell shaped distribution of
scores.
Example:
Scores: 7, 8, 9, 9, 10, 10, 10, 11, 11, 12, 13
Mean: 10
Median: 10
Mode: 10
Range: 6

38. Scores that are "bunched" at the right or high end of the scale
are said to have a negative skew.
 If you have data where the mean, median and mode
are quite different, the scores are said to be skewed.
Example:
Scores: 7, 8, 9, 10, 11, 11, 12, 12, 12, 13, 13
Mean: 10.7
Median: 11
Mode: 12
Range: 6

39. Negatively Skewed
Scores that are "bunched" at the right or high end of
the scale are said to have a negative skew.

40. Positively Skewed
In a positively skewed curve scores are
bunched near the left or low end of a
scale.

41. Question: The survey of the students in three
classes showed differences in how long the
students studied for each course. The mean
number of hours for students in Course A was
about _____, and for students in Courses B
and C, the average was about ________. Does
this mean Course A requires the most hours
of study? Were the differences the
researcher observed in study time real or just
due to chance? In other words, can she
generalize from the samples of students she
surveyed to the whole population of
students? She needs to determine the
reliability and significance of her statistics.

42. References
 Review Information at these links:
 http://www.psychstat.missouristate.edu/introb
ook/sbk11.htm
 http://www.socialresearchmethods.net/kb/meas