1. Co03.jpg

2. Overview 3.1 Measures of Center 3.2 Measures of Variability
3.4 Measures of Position
3.6 Robust Measures

3. 3.1 Measures of Center Objectives:
By the end of this section, I will be
able to…
Calculate the mean for a given data set.
Find the median, and describe why the median is sometimes preferable to the mean.
Find the mode of a data set.
Describe how skewness affects these measures of center.

4. The Mean Most well known and widely used measure of center
Simply add up all the numbers
and divide by how many numbers you have.

5. Notation Statisticians like to use specialized notation.
Sample size - how many observations you have in your sample data set, is always denoted by n
ith data value by xi, where i is simply an index or counter indicating a data point
“add them together” is Σ (capital sigma)
The sample mean is called (pronounced “x-bar”)

6. The sample mean
Written as In plain English, this just means that, in order to find the mean x, we 1. Add up all the data values, giving us Σx. 2. And divide by how many observations are in the data set, giving us Σx /n.

7. The Population Mean m Mean value of a population is usually unknown
Use x to estimate m
Denote the population mean with m (mu)
Population size is denoted by N.
The mean is sensitive to the presence of extreme values

8. The Median
The middle data value when the data are put into ascending order
Half of the data values lie below the median and half lie above
If the sample size n is odd, then the median is a unique middle value.
That is, observation when the data are put in
ascending order.
If the sample size n is even, then the median is the mean of the two data values in the middle.
That is, the median is the mean of the two data
values that lie on either side of the position.

9. The Mode
French speakers will recognize that the term mode in French refers to fashion
The popularity of clothing often depends on just which style is in fashion
In a data set, the value that is most “in fashion” is the value that occurs the most
The mode of a data set is the data value that occurs with the greatest frequency

10. Example 3.5 - Cost of mathematical journals
The rising cost of research journals has been taking an increasing bite out of library and research budgets. Table 3.3 contains the annual subscription cost of ten research journals in mathematics and statistics for Find the following. a. The mean journal subscription cost b. The median journal subscription cost c. The mode journal subscription cost

11. Table 3.3 Annual subscription cost for ten research journals
Example 3.5 continued
Table 3.3 Annual subscription cost for ten research journals

12. Example 3.5 continued
Solution a. The sample mean journal cost is

13. Example 3.5 continued
b. Since we have n 10 journals, the median is the mean of the two data values that lie on either side of the The median is the mean of the 5th and 6th data values, $850 and $1022 median journal cost =

14. Example 3.5 continued c. The mode is the data value that occurs with
the greatest frequency.
Only two journals that cost $250 each.
No other cost occurs more than once.
Mode = $250.
Mode is not a very good measure of center for this data set because it is the minimum value.
Illustrates a weakness of using the mode as a measure of center.

15. How Skewness Affects the Mean and Median
For a right-skewed distribution, the mean is larger than the median.
For a left-skewed distribution, the median is larger than the mean.
For a symmetric unimodal distribution, the mean, median, and mode are fairly close to one another.
FIGURE 3.5 How skewness affects the mean and median.

16. Exploratory Data Analysis
Using graphical methods to compare numerical statistics
FIGURE 3.6 Dotplots of the percentage net price change for the Dow Jones Industrial Average, the randomly selected darts portfolio, and the professionally selected portfolio.

17. Summary
The sample mean represents the sum of the data values in the sample divided by the sample size (n).
The population mean (m) represents the sum of the data values in the population divided by the population size (N).
The mean is sensitive to the presence of extreme values.

18. Summary
The median occupies the middle position when the data are put in ascending order and is not sensitive to extreme values.
The mode is the data value that occurs with the greatest frequency.
Modes can be applied to categorical data as well as numerical data but are not always reliable as measures of center.

19. Summary
The skewness of a distribution can often tell us something about the relative values of the mean and the median.

20. 3.2 Measures of Variability
Objectives:
By the end of this section, I will be
able to…
Understand and calculate the range of a data set.
Explain in my own words what a deviation is.
Calculate the variance and the standard deviation for a population or a sample.

21. The Range
The difference between the largest value and the smallest value in the data set:
range = largest value – smallest value
Simplest measure of variability
Larger range is an indication of greater variability

22. Example 3.8 - Range of the volleyball teams’ heights
Calculate the range of player heights for each of the WMU and NCU teams.
FIGURE 3.11 Comparative dotplots of the heights of two volleyball teams.

23. Example 3.8 continued Solution
From Figure 3.11shows WMU heights are more spread out than NCU heights.
Range of WMU team should be larger than the range of the NCU team, reflecting greater variability.
rangeWMU = = 15 inches
rangeNCU = 72 – 66 = 6 inches

24. What Is a Deviation?
A deviation for a given data value x is the difference between the data value and the mean of the data set.
For a sample, the deviation equals x - x.
For a population, the deviation equals x - m.
Data value is larger than the mean, the deviation will be positive

25. Deviation
Data value is smaller than the mean, the deviation will be negative
Data value equals the mean, the deviation will be zero
Deviation can roughly be thought of as the distance between a data value and the mean
The deviation can be negative while distance is always positive
Deviation not useful measure of spread because sum of deviations is always zero.

26. Population Variance s2
Symbolized by the lowercase Greek letter sigma squared, s2
Is the mean of the squared deviations in the population and is found by

27. The Population Standard Deviation s
The square root of the variance
Represents a distance from the mean that is representative for that data set
Not the mean deviation, which is always zero

28. Sample Variance s2
Based on the idea of finding the sum of the squared deviations S(x – x)2 and then dividing by the sample size to get the mean squared deviation
Statisticians found a better estimate by dividing by n - 1

29. Sample Standard Deviation s
The square root of the sample variance s2
Second most important statistic
The value of s may be interpreted as the typical difference between a data value and the sample mean

30. Computational Formulas
Population variance:
Population standard deviation:
Sample variance:
Sample standard deviation:

31. Example Calculating the population variance and population standard deviation using the calculator.
Table 3.13 lists the amount of farmland (in 1000s of acres) in each county in the state of Connecticut. Since the data set contains all N = 8 counties in Connecticut, it can be considered a population. Calculate the population variance and population standard deviation using the calculator.

32. Table 3.13 Farmland in Connecticut
Example 3.15 continued
Table 3.13 Farmland in Connecticut

33. Example 3.15 continued The population standard deviation is therefore:
The standard deviation of farmland for all counties in Connecticut is almost 25,100 acres.

34. Summary
The simplest measure of variability, or measure of spread, is the range.
The range is simply the difference between the maximum and minimum values in a data set
The range has drawbacks because it relies on the two most extreme data values.
A deviation is the difference between a data value and the mean of the data values.

35. Summary
The variance and standard deviation are measures of spread that utilize all available data values.
The population variance can be thought of as the mean squared deviation.
The standard deviation is the square root of the variance.
Standard deviation is a typical deviation, that is, the typical difference between a data value and the mean.

36. 3.4 Measures of Position Objectives:
By the end of this section, I will be
able to…
Find percentiles for both small and large data sets.

37. Percentile Let p be any integer between 0 and 100.
The pth percentile of a data set is the data value at which p percent of the values in the data set are less than or equal to this value.

38. Example 3.24 - Finding percentiles of a small data set
Yolanda would like to go to a prestigious graduate school of the arts. She knows that this school accepts only those students who score at the 75th percentile or higher in a grueling dance audition. The following data represent the dance audition scores of Yolanda’s group. Yolanda scored 85. Find the 75th percentile of the data set. Will Yolanda be accepted at the prestigious graduate school of the arts?

39. Example 3.24 continued
Solution Step 1: Sort the data into ascending order Step 2: Since we want the 75th percentile, p=75. There are 12 scores, so n=12. Calculate So, i = 9.

40. Example 3.24 continued Step 3:
Here, since i is an integer, the 75th percentile
is the mean of the data values in positions 9
and 10.
Data value in the ninth position is 81.
Data value in the tenth position is 85.
Mean of these values is 83. Thus, the 75th percentile is 83.
Yolanda’s dance score of 85 is therefore
above the 75th percentile. She will be
accepted to the prestigious graduate school.

41. Outliers
Extremely large or extremely small data value relative to the rest of the data set
May represent a data entry error, or it may be genuine data
Farther than three standard deviations from the mean

42. Summary
Measures of position, which tell us the position that a particular data value holds relative to the rest of the data set.
The pth percentile of a dataset is the value at
which p percent of the values in the data
set are less than or equal to this value.

43. 3.6 Robust Measures Objectives: By the end of this section, I will be
able to…
Find quartiles and the interquartile range.
Calculate the five-number summary of a data set.
Construct a boxplot for a given data set.
Apply robust detection of outliers.

44. Quartiles
Divide the data set into quarters
FIGURE 3.31

45. Quartiles Each part contains 25% of the data.
The first quartile (Q1) is the 25th percentile.
The second quartile (Q2) is the 50th percentile, that is, the median.
The third quartile (Q3) is the 75th percentile.
For small data sets, the division may be into four parts of only approximately equal size.

46. Example 3.35 - Finding the quartiles for a small data set: the dance audition scores
In Example 3.24 (page 126) we examined the dance scores of 12 students auditioning for admission into a prestigious graduate school of the arts. Recall that we found the 75th percentile of the dance audition scores to be 83. By definition, the 75th percentile is the third quartile Q3. Therefore, this score of 83 is also the third quartile (Q3) of the audition scores. Now we will find the first quartile and the median (second quartile).

47. FIGURE 3.34 The quartiles for the dance audition data.
Example 3.35 continued
FIGURE 3.34 The quartiles for the dance audition data.

48. Interquartile Range
Interquartile range (IQR) is a robust measure of variability.
IQR = Q3 - Q1
The interquartile range is interpreted to be the spread of the middle 50% of the data.

49. Example 3.37 - Finding the interquartile range for the dance audition scores
In Example 3.35, we found that, for the dance audition score data, Q1 = 59 and Q3 = 83. Find the IQR for the dance score data and explain what it means.

50. FIGURE 3.38 The IQR for the dance audition data.
Example 3.37 continued
Solution
We would say that the middle 50%, or middle half, of the dance audition scores ranged over 24 points (see Figure 3.38).
FIGURE 3.38 The IQR for the dance audition data.

51. Example 3.37 continued
What would happen if we introduced an outlier into this data set?
Change the lowest score from 30 to 3?
IQR completely unaffected even if we changed the 44 to a 4.
f we changed the 56, then the IQR would be affected, since Q1 would then change.

52. The Five-Number Summary
Consists of the following set of statistics, which together constitute a robust summarization of a data set: 1. Smallest value in the data set (minimum) 2. First quartile, Q1 3. Median, Q2 4. Third quartile, Q3 5. Largest value in the data set (maximum)

53. FIGURE 3.39 The quartiles for the dance audition data.
Example Find the five-number summary for the dance audition data.
Solution: Figure 3.39 shows the five-number summary
FIGURE 3.39 The quartiles for the dance audition data.

54. Example 3.38 continued
1. Minimum = First quartile, Q1 = Median Q2 = Third quartile, Q3 = Maximum = 94

55. Example 3.38 continued
The five-number summary is often reported as Min = 30, Q1 = 59, Med = 71.5, Q3 = 83, Max = 94.
Which parts of the five-number summary are less robust than others?
Since the minimum and maximum are the most extreme values, these are clearly very sensitive to outliers.
However, Q1, the median, and Q3 are very
resistant to the influence of outliers.

56. The Boxplot Convenient graphical display of the five- number summary
Allows the data analyst to evaluate the symmetry or skewness of a data set.

57. Example 3.41 - Constructing a Boxplot by hand
Construct a boxplot for the dance score data. Use the steps for constructing a boxplot on page 148.

58. Example 3.41 continued Solution
From Example 3.38, the five-number summary for the dance score data is
Min = 30, Q1 = 59, Med = 71.5, Q3 = 83, Max = 94.
From Example 3.37, the interquartile
range for the dance score data is
IQR = Q3 - Q1 = = 24.
Step 1: Determine the lower and upper fences:
a. Lower fence Q (IQR)
= (24) = = 23
b. Upper fence Q (IQR)
= (24) = = 119

59. Example 3.41 continued Step 2: Draw a horizontal number line that
encompasses the range of your data,
including the fences.
Above the number line, draw vertical lines at Q1 = 59, median = 71.5, and Q3 = 83.
Connect the lines for Q1 and Q3 to each other so as to form a box, as shown in Figure 3.41a below.

60. Example 3.41 continued
Step 3: Temporarily indicate the fences (lower fence 23 and upper fence 119) as brackets above the number line. (See Figure 3.41b below.)

61. Example 3.41 continued Step 4:
Draw a horizontal line from Q1 = 59 to the
smallest data value greater than the lower
fence.
The lowest data value is Min = 30.
This is greater than the lower fence = 23.
So draw the line from 59 to 23.
Draw a horizontal line from Q3 = 83 to the largest data value smaller than the upper fence.

62. Example 3.41 continued
The largest data value is Max = 94, which is smaller than the upper fence.
So draw the line from 83 to 94.
(See Figure 3.41c below.)

63. Example 3.41 continued Step 5:
There are no data values lower than the lower
fence or greater than the upper fence.
No outliers in this data set.
Remove the temporary brackets.
See Figure 3.41d below.

64. Robust Detection of Outliers
Use a five-number summary or a boxplot to
detect outliers, as follows:
A data value is an outlier if
It is located 1.5(IQR) or more below Q1, or
It is located 1.5(IQR) or more above Q3.

65. Example 3.45 - Robust detection of outliers for the dance audition data
Determine if there are any outliers in the dance score data.

66. Example 3.45 continued Solution
Recall for the dance score data set that
IQR = 24, Q1 = 59, and Q3 = 83.
So we have 1.5(IQR) = 1.5(24) = 36.
Q1 – 1.5(IQR) and Q (IQR):
The first step is to find the two quantities
Q1 – 1.5(IQR) = Q1 – 36 = 59 – 36 = 23
Q (IQR) = Q = = 119

67. Example 3.45 continued
No data values less then or equal to 23 or greater than or equal to 119.
No outliers are identified by the robust method.

68. Summary
Section 3.6 presents robust measures and methods, which are not sensitive to the presence of outliers.
Quartiles divide the data set into approximately equal quarters.
The interquartile range is a measure of variability found by taking the difference between the third and first quartiles.

69. Summary The five-number summary is a robust
alternative to the usual mean-and-standard-
deviation method of summarizing a data set.
It consists of simply reporting the minimum, first quartile, median, third quartile, and maximum of the data set.

70. Summary
A boxplot is a graphical representation of the five-number summary and is useful for investigating skewness and the presence of outliers.
The robust method of detecting outliers is to consider a data value an outlier if it is located 1.5(IQR) or more below Q1, or it is located 1.5(IQR) or more above Q3.