Lecture discussing the measures of central tendency

1. Introduction to statistics-3
Measures of variability or spread

2. Summary Last Class
• Measures of central Tendency
– Mean
– Median
– Mode

3. Mode
• The mode can be found with any scale of
measurement; it is the only measure of
typicality that can be used with a nominal
scale.

4. Median
• The median can be used with ordinal, as well
as interval/ratio, scales. It can even be used
with scales that have open-ended categories
at either end (e.g., 10 or more).
• It is not greatly affected by outliers, and it can
be a good descriptive statistic for a strongly
skewed distribution.

5. Mean
• The mean can only be used with interval or
ratio scales. It is affected by every score in the
distribution, and it can be strongly affected by
outliers.
• It may not be a good descriptive statistic for a
skewed distribution, but it plays an important
role in advanced statistics.

6. Key Terms
• Variability
• Measures of Variability
• Variance
• Standard Deviation
• Normal Distribution

7. Variability
• The difference in data or in a set of scores
• Populations can be described as homogenous
or heterogeneous based on the level of
variability in the data set

8. Measures of Variability
• Provide an estimate of how much scores in a
distribution vary from an average score. The
usual average is the mean

9. • Range
• Variance
• Standard Deviation

10. Range
• A single number that represents the spread of
the data
• Upper real limit of Xmax- Lower real limit of
Xmin

11. Variance
• Variance is a measure of variability
• Standard deviation is the square root of
variance

12. Steps to calculate SD
1. Determine the Mean= x̄
2. Determine the deviations (X-x̄)
3. Square these (X-x̄)2
4. Add the squares ∑ (X-x̄)2

13. 5. Divide by total numbers less one ∑ (X-x̄)2/n-1
6. Square root of result is Standard Deviation
Standard Deviation=√∑ (X-x̄)2/n-1

14. Variance and Standard Deviation
Population Sample
Mean =µ Mean= x̄
Variance Variance
Standard Deviation Standard Deviation

15. Worked Example (appspot.com)

16. Step 1- Calculate mean

17. Step 2 Calculate Deviation

18. Step 3 Sum Mean Square Deviation

19. Step 4 Calculate standard deviation

21. Range
• The range tells you the largest difference that
you have among your scores. It is strongly
affected by outliers, and being based on only
two scores, it can be very unreliable.

22. Mean Deviation
• The mean deviation, and the two measures
that follow, can only be used with
interval/ratio scales.
• It is a good descriptive measure, which is less
affected by outliers than the standard
deviation, but it is not used in advanced
statistics.

23. Variance
• The variance is not appropriate for descriptive
purposes, but it plays an important role in
advanced statistics.

24. Standard Deviation
• The standard deviation is a good descriptive
measure of variability, although it can be
affected strongly by outliers. It plays an
important role in advanced statistics.

25. Scaled Scores
• A key element of statistics is making
comparison between variables and
populations.

26. Changing the mean and standard
deviation of a distribution
• If a constant is added to each score in a
distribution the mean for the distribution
changes but the variance and standard
deviation does not.
• Adding a score changes the sum of all the
scores but not the spread or shape of a
distribution

27. Adding or Subtracting a constant
• When you add or subtract a constant from
each score in a distribution the mean changes
by the amount added or subtracted but the
standard deviation and variance remain the
same
x̄ new= x̄ original +/- constant
s new= s original

28. Multiplying or dividing by a constant
x̄ new= x̄ original x or / by the constant
s new= s original x or / by the constant

29. Z scores
• The z score provides the exact position of a
score in its distribution.
• This allows us to compare scores from
different distributions

30. Z-score distribution
• The z score distribution has a mean of 0 and a
standard deviation of 1

31. Z-Score formula

32. Z score for sample
Xi X
z
S

33. Example Z Score
• For scores above the mean, the z score has a
positive sign. Example + 1.5z.
• Below the mean, the z score has a minus sign.
Example - 0.5z.
• Calculate Z score for blood pressure of 140 if
the sample mean is 110 and the standard
deviation is 10
• Z = 140 – 110 / 10 = 3
33

34. Comparing Scores from Different
Distributions
• Interpreting a raw score requires additional
information about the entire distribution. In most
situations, we need some idea about the mean score
and an indication of how much the scores vary.
• For example, assume that an individual took two
tests in reading and mathematics. The reading score
was 32 and mathematics was 48. Is it correct to say
that performance in mathematics was better than in
reading?
34

35. Z Scores Help in Comparisons
• Not without additional information. One
method to interpret the raw score is to
transform it to a z score.
• The advantage of the z score transformation is
that it takes into account both the mean value
and the variability in a set of raw scores.
35

36. Example 1
Dave in Statistics:
15
Statistics Calculus (50 - 40)/10 = 1
(one SD above the
mean)
10
Dave in Calculus
5
(50 - 60)/10 = -1
(one SD below the
mean)
0
0 20406080100
Mean Statistics G RADE
Mean
= 40 Calculus = 60

37. Example 2
An example where the
means are identical, but
0 5 10 15 20 25 30
the two sets of scores
Statistics have different spreads
Dave’s Stats Z-score
(50-40)/5 = 2
Calculus Dave’s Calc Z-score
(50-40)/20 = .5
0 20406080100
G RA DE

38. Thee Properties of Standard Scores
• 1. The mean of a set of z-scores is always
zero

39. Properties of Standard Scores
• Why?
• The mean has been subtracted from each
score. Therefore, following the definition
of the mean as a balancing point, the sum
(and, accordingly, the average) of all the
deviation scores must be zero.

40. Three Properties of Standard Scores
• 2. The SD of a set of standardized scores is
always 1

41. Three Properties of Standard Scores
• 3. The distribution of a set of standardized
scores has the same shape as the
unstandardized scores
– beware of the “normalization”
misinterpretation

42. The shape is the same
(but the scaling or metric is different)
UNSTANDARDIZED STANDARDIZED
0.5
6
0.4
4
0.3
0.2
2
0.1
0.0
0
0.4 0.6 0.8 1.0 -6 -4 -2 0 2

43. Two Advantages of Standard Scores
1. We can use standard scores to find centile
scores: the proportion of people with scores
less than or equal to a particular score.
Centile scores are intuitive ways of
summarizing a person’s location in a larger set
of scores.

44. The area under a normal curve
0. 0.1 0.2 0.3 0.4 50%
34% 34%
14% 14%
2% 2%
-4 -2 0 2 4
SCO RE

45. Two Advantages of Standard Scores
2. Standard scores provides a way to standardize
or equate different metrics. We can now
interpret Dave’s scores in Statistics and
Calculus on the same metric (the z-score
metric). (Each score comes from a distribution
with the same mean [zero] and the same
standard deviation [1].)

46. Two Disadvantages of Standard Scores
1. Because a person’s score is expressed relative to
the group (X - M), the same person can have
different z-scores when assessed in different
samples
Example: If Dave had taken his Calculus exam in a
class in which everyone knew math well his z-
score would be well below the mean. If the class
didn’t know math very well, however, Dave
would be above the mean. Dave’s score depends
on everyone else’s scores.

47. Two Disadvantages of Standard Scores
2. If the absolute score is meaningful or of
psychological interest, it will be obscured by
transforming it to a relative metric.

48. Properties of the Mean, Standard
Deviation, and Standardized Scores
Mean. Adding or subtracting a constant from
the scores changes the mean in the same way.
Multiplying or dividing by a constant also
changes the mean in the same way. The sum of
squared deviations is smaller around the mean
than any other point in the distribution.

49. Standard Deviation
• Standard deviation. Adding or subtracting a
constant from the scores does not change the
standard deviation. However, multiplying or
dividing by a constant means that the
standard deviation will be multiplied or
divided by the same constant. The standard
deviation is smaller when calculated around
the mean than any other point in the
distribution.

50. Standardized Scores
• Standardized scores. Adding, subtracting,
multiplying or dividing the scores by a
constant does not change the standardized
scores. The mean of a set of z scores is zero,
and the standard deviation is 1.0.