1. 2.5: Numerical Measures of Variability (Spread)
The mean, median and mode give us an idea of the central tendency, or where the “middle” of the data is.
Variability gives us an idea of how spread out the data are around that middle.
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

2. 2.4: Numerical Measures of Variability
Range
Variance
Standard Deviation (SD)
IQR (interquartile range)
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

3. Range
The range is equal to the largest value minus the smallest value in the data set.
Easy to compute, but not very informative.
Considers only two observations (the smallest and largest).
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

4. Range
Ex: Data (Number of Vacation Days in a Company): 22, 17, 15, 16, 14, 20, 25, 11, 26, 14, 23, 21, 13, 15, 15, 28, 30, 20, 14, 33, 27, 28, 15, 22, 16, 12, 25, 31, 19, 23, 26, 21, 11, 18, 17
Range=largest-smallest=33-11=22
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

5. Sample Variance
The sample variance, s2, for a sample of n measurements is equal to the sum of the squared distances from the mean, divided by (n – 1).
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

6. Sample Standard Deviation (SD)
The sample standard deviation, s, for a sample of n measurements is equal to the square root of the sample variance.
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

7. Example 1:
Say a small data set consists of the measurements 1, 3, 5, and 3.
(when we have few numbers definition and calculation formula take about the same time, but for data with many numbers calculation formula is easier to use!)
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

8. Example 2:
Ex: Data (Number of Vacation Days in a Company): 22, 17, 15, 16, 14, 20, 25, 11, 26, 14, 23, 21, 13, 15, 15, 28, 30, 20, 14, 33, 27, 28, 15, 22, 16, 12, 25, 31, 19, 23, 26, 21, 11, 18, 17
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

9. 2.5: Numerical Measures of Variability
As before, Greek letters are used for populations and Roman letters for samples:
s2 = sample variance
s = sample standard deviation
s2 = population variance
s = population standard deviation
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

10. 2.6: Interpreting the Standard Deviation
Chebyshev’s Rule
The Empirical Rule
Both tell us something about where
the data will be relative to the mean.
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

11. Chebyshev’s Rule Chebyshev’s Rule k k2 1/ k2 2 4 .25 75% 3 9 .11 89%
Valid for any data set
For any number k >1, at least (1-1/k2) ×100% of the observations will lie within k standard deviations of the mean. k k2
1/ k2
(1- 1/ k2 ) ×100% 2 4
.25
75% 3 9
.11
89% 16
.0625
93.75%
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

12. Chebyshev’s Rule
Ex: Data (Number of Vacation Days in a Company): 22, 17, 15, 16, 14, 20, 25, 11, 26, 14, 23, 21, 13, 15, 15, 28, 30, 20, 14, 33, 27, 28, 15, 22, 16, 12, 25, 31, 19, 23, 26, 21, 11, 18, 17
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

13. Chebyshev’s Rule
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

14. Empirical Rule
For a perfectly symmetrical and bell-shaped distribution,
~68% will be within the range
~95% will be within the range
~99.7% will be within the range
The Empirical Rule
Useful for bell-shaped, symmetrical distributions
If not perfectly bell-shaped and symmetrical, the values are approximations.
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

15. Empirical Rule
Ex: Data (Number of Vacation Days in a Company): 22, 17, 15, 16, 14, 20, 25, 11, 26, 14, 23, 21, 13, 15, 15, 28, 30, 20, 14, 33, 27, 28, 15, 22, 16, 12, 25, 31, 19, 23, 26, 21, 11, 18, 17
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

16. Empirical Rule
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

17. Empirical Rule (Example 3)
Hummingbirds beat their wings in flight an average of 55 times per second.
Assume the standard deviation is 10, and that the distribution is symmetrical and bell-shaped.
Approximately what percentage of hummingbirds beat their wings between 45 and 65 times per second?
Between 55 and 65?
Less than 45?
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

18. Empirical Rule (Example 3)
Since 45 and 65 are exactly one standard deviation below and above the mean, the empirical rule says that about 68% of the hummingbirds will be in this range.
Hummingbirds beat their wings in flight an average of 55 times per second.
Assume the standard deviation is 10, and that the distribution is symmetrical and bell-shaped.
Approximately what percentage of hummingbirds beat their wings between 45 and 65 times per second?
Between 55 and 65?
Less than 45?
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

19. Empirical Rule (Example 3)
Hummingbirds beat their wings in flight an average of 55 times per second.
Assume the standard deviation is 10, and that the distribution is symmetrical and bell-shaped.
Approximately what percentage of hummingbirds beat their wings between 45 and 65 times per second?
Between 55 and 65?
Less than 45?
This range of numbers is from the mean to one standard deviation above it, or one-half of the range in the previous question. So, about one-half of 68%, or 34%, of the hummingbirds will be in this range.
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

20. Empirical Rule (Example 3)
Hummingbirds beat their wings in flight an average of 55 times per second.
Assume the standard deviation is 10, and that the distribution is symmetrical and bell-shaped.
Approximately what percentage of hummingbirds beat their wings between 45 and 65 times per second?
Between 55 and 65?
Less than 45?
Half of the entire data set lies above the mean, and ~34% lie between 45 and 55 (between one standard deviation below the mean and the mean), so by symmetry ~34% lie between 45 and 55, which means ~16% are below 45.
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

21. 2.7: Numerical Measures of Relative Standing:Percentiles
Percentiles: for any (large) set of n measurements (arranged in ascending order), the 100×pth percentile is a number such that at least 100p% of the measurements fall at or below that number and at least 100(1 – p)% fall at or above it.
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

22. Percentiles
Finding percentiles is similar to finding the median – the median is the 50th percentile.
If you are in the 50th percentile for the GRE, half of the test-takers scored like you or better and half scored like you or worse.
If you are in the 75th percentile, three-quarters of the test-takers scored like you or worse.
If you are in the 90th percentile, only 10% of all the test-takers scored like you or better.
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

23. Finding Percentiles 1. Order the data from smallest to largest
2. Find k=n×p.
(a) If k is an integer, 100×pth percentile is the average of the kth and (k+1)th values.
(b) If k is not an integer, round it up to the next integer, say q. Then 100×pth percentile is the qth value.
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

24. Finding Percentiles
Ex: Data (Number of Vacation Days in a Company): Ordered: 11, 11, 12, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 17, 17, 18, 19, 20, 20, 21, 21, 22, 22, 23, 23, 25, 25, 26, 26, 27, 28, 28, 30, 31, 33
(i) Find the 80th percentile.
k=n×p=35×0.80=28 is an integer, so 80th percentile is the average of 28th and 29th values,
i.e., 80th percentile = (26+26)/2=26
Check: 29 out of 35 values (i.e., ~83% of the data) are ≤26 (which satisfies at least 80%!)
8 out of 35 values (i.e., ~23% of the data) are ≥26 (which satisfies at least 20%!)
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

25. Finding Percentiles
(ii) Find the 25th percentile (first quartile or lower quartile, QL).
k=n×p=35×0.25=8.75 is not an integer, so 25th percentile is 9th value,
i.e., 25th percentile = 15
(ii) Find the 75th percentile (third quartile or upper quartile, QU).
k=n×p=35×0.75=26.25 is not an integer, so 75th percentile is 27th value,
i.e., 75th percentile = 25
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

26. Interquartile Range (IQR)
IQR is another measure of spread or variability!
It is the difference between third quartile and first quartile,
That is,
IQR=75th percentile minus 25th percentile= QU -QL
Ex: in the number of vacation days of 35 employees
IQR=25-15=10
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

27. 2.7: Numerical Measures of Relative Standing: z-score
The z-score tells us how many standard deviations above or below the mean a particular measurement is.
Sample z-score
Population z-score
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

28. 2.6: Interpreting the Standard Deviation
Hummingbirds beat their wings in flight an average of 55 times per second.
Assume the standard deviation is 10, and that the distribution is symmetrical and bell-shaped.
An individual hummingbird is measured with 75 beats per second. What is this bird’s z-score?
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

29. 2.6: Interpreting the Standard Deviation
Since ~95% of all the measurements will be within 2 standard deviations of the mean, only ~5% will be more than 2 standard deviations from the mean.
About half of this 5% will be far below the mean, leaving only about 2.5% of the measurements at least 2 standard deviations above the mean.
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

30. 2.7: Numerical Measures of Relative Standing
Z scores are related to the empirical rule:
For a perfectly symmetrical and bell-shaped distribution,
~68 % will have z-scores between -1 and 1
~95 % will have z-scores between -2 and 2
~99.7% will have z-scores between -3 and 3
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

31. 2.8: Methods for Determining Outliers
An outlier is a measurement that is unusually large or small relative to the other values.
Three possible causes:
Observation, recording or data entry error
Item is from a different population
A rare, chance event
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

32. 2.8: Methods for Determining Outliers
The box plot is a graph representing information about certain percentiles for a data set and can be used to identify outliers
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

33. 2.8: Methods for Determining Outliers
Lower Quartile
(QL)
Median
Upper Quartile
(QU)
Minimum Value
inside the inner fence
Maximum Value
inside the inner fence
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

34. 2.8: Methods for Determining Outliers
Interquartile Range (IQR) = QU - QL
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

35. 2.8: Methods for Determining Outliers
Right Outer Fence = QU + 3(IQR)
Right Inner Fence = QU + 1.5(IQR)
Left Inner Fence = QL - 1.5(IQR) and
Left Outer Fence = QL – 3(IQR)
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

36. 2.8: Methods for Determining Outliers
Outliers and z-scores
The chance that a z-score is between -3 and +3 is over 99%.
Any measurement with |z| > 3 is considered an outlier.
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

37. 2.8: Methods for Determining Outliers
Outliers and z-scores
Here are the descriptive statistics for the games won at the All-Star break, except one team had its total wins for 2006 recorded.
That team, with 104 wins recorded, had a z-score of ( )/12.11 = 4.82.
That’s a very unlikely result, which isn’t surprising given what we know about the observation.
# of Wins
n = 30
Mean
45.68
Sample Variance
146.69
Sample Standard Deviation
12.11
Minimum 25
Maximum
104
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

38. Robustness to Outliers
Ex: Consider the data: 2, 7, 5, 6, 4, 2, 5, 1, 5, 6
Mean=4.3, Median=5, Mode=5
Range=6, Variance=4.01, SD=2.0, IQR=3.25
Ex: what if data were 2, 7, 5, 6, 4, 2, 5, 1, 5, 100, then
Mean=13.7, Median=5, Mode=5
Range=99, Variance=923.12, SD=30.38, IQR=3.25
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

39. Robustness to Outliers
Hence, mean, range, variance, and standard deviation are highly affected by the outliers (or extreme values)
While, median, mode, and IQR are not affected by the outliers, i.e., they are robust to outliers.
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

40. 2.9: Graphing Bivariate Relationships
Scattergram (or scatterplot) shows the relationship between two quantitative variables
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

41. 2.9: Graphing Bivariate Relationships
If there is no linear relationship between the variables, the scatterplot may look like a cloud, a horizontal line or a more complex curve
Source: Quantitative Environmental Learning Project
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data

42. 2.10: Distorting the Truth with Deceptive Statistics
Distortions
Stretching the axis (and the truth)
Is average center?
Mean, median or mode?
Is average relevant?
What about the spread?
McClave, Statistics , 11th ed. Chapter 2: Methods for Describing Sets of Data