1. Chapter 2 ~ Descriptive Analysis & Presentation of Single-Variable Data
Mean: inches
Median: inches
Range: 42 inches
Variance:
Standard deviation: inches
Minimum: 36 inches
Maximum: 78 inches
First quartile: inches
Third quartile: inches
Count: 58 bears
Sum: inches 10 20 30 40 50 60 70 80
Frequency
Length in Inches
Black Bears

2. Chapter Goals Learn how to present and describe sets of data
Learn measures of central tendency, measures of dispersion (spread), measures of position, and types of distributions
Learn how to interpret findings so that we know what the data is telling us about the sampled population

3. 2.1 ~ Graphic Presentation of Data
Use initial exploratory data-analysis techniques to produce a pictorial representation of the data
Resulting displays reveal patterns of behavior of the variable being studied
The method used is determined by the type of data and the idea to be presented
No single correct answer when constructing a graphic display

4. Circle Graphs & Bar Graphs
Circle Graphs and Bar Graphs: Graphs that are used to summarize attribute data
Circle graphs (pie diagrams) show the amount of data that belongs to each category as a proportional part of a circle
Bar graphs show the amount of data that belongs to each category as proportionally sized rectangular areas

5. Example
Example: The table below lists the number of automobiles sold last week by day for a local dealership. Describe the data using a circle graph and a bar graph:
Day
Number Sold
Monday 15
Tuesday 23
Wednesday 35
Thursday 11
Friday 12
Saturday 42

6. Circle Graph Solution
Automobiles Sold Last Week

7. Bar Graph Solution
Automobiles Sold Last Week
Frequency

8. Pareto Diagram
Pareto Diagram: A bar graph with the bars arranged from the most numerous category to the least numerous category. It includes a line graph displaying the cumulative percentages and counts for the bars.
Notes:
Used to identify the number and type of defects that happen within a product or service
Separates the “vital few” from the “trivial many”
The Pareto diagram is often used in quality control applications

9. Example
Example: The final daily inspection defect report for a cabinet manufacturer is given in the table below:
Defect
Number
Dent 5
Stain 12
Blemish 43
Chip 25
Scratch 40
Others 10
1) Construct a Pareto diagram for this defect report. Management has given the cabinet production line the goal of reducing their defects by 50%.
2) What two defects should they give special attention to in working toward this goal?

10. Daily Defect Inspection Report
Solutions 1 4 2 8 6
Count
Percent
Blemish
Scratch
Chip
Stain
Others
Dent
Defect: 43 40 25 12 10 5
31.9
29.6
18.5
8.9
7.4
3.7
Cum%
61.5
80.0
88.9
96.3
100.0
Daily Defect Inspection Report 1)
2) The production line should try to eliminate blemishes and scratches. This would cut defects by more than 50%.

11. Key Definitions
Quantitative Data: One reason for constructing a graph of quantitative data is to examine the distribution - is the data compact, spread out, skewed, symmetric, etc.
Distribution: The pattern of variability displayed by the data of a variable. The distribution displays the frequency of each value of the variable.
Dotplot Display: Displays the data of a sample by representing each piece of data with a dot positioned along a scale. This scale can be either horizontal or vertical. The frequency of the values is represented along the other scale.

12. Example : . . .:. . ..: :.::::::.. .::. ... . : . . . :. . .
Example: A random sample of the lifetime (in years) of 50 home washing machines is given below:
2.5
8.9
12.2
4.1
18.1
1.6
16.9
3.5
0.4
2.6
2.2
4.0
4.5
6.4
2.9
3.3
4.4
9.2
0.9
14.5
7.2
5.2
1.8
1.5
0.7
3.7
4.2
6.9
15.3
21.8
17.8
7.3
6.8
7.0
18.3
8.5
1.4
7.4
4.7
10.4
3.6
The figure below is a dotplot for the 50 lifetimes: .
: :
..: :.::::::.. .:: : :
Note: Notice how the data is “bunched” near the lower extreme and more “spread out” near the higher extreme

13. Stem & Leaf Display Background:
The stem-and-leaf display has become very popular for summarizing numerical data
It is a combination of graphing and sorting
The actual data is part of the graph
Well-suited for computers
Stem-and-Leaf Display: Pictures the data of a sample using the actual digits that make up the data values. Each numerical data is divided into two parts: The leading digit(s) becomes the stem, and the trailing digit(s) becomes the leaf. The stems are located along the main axis, and a leaf for each piece of data is located so as to display the distribution of the data.

14. Example
Example: A city police officer, using radar, checked the speed of cars as they were traveling down the main street in town. Construct a stem-and-leaf plot for this data:
Solution:
All the speeds are in the 10s, 20s, 30s, 40s, and 50s. Use the first digit of each speed as the stem and the second digit as the leaf. Draw a vertical line and list the stems, in order to the left of the line. Place each leaf on its stem: place the trailing digit on the right side of the vertical line opposite its corresponding leading digit.

15. Example 20 Speeds --------------------------------------- 1 | 6 9
1 | 6 9
2 |
3 |
4 |
5 | 5
The speeds are centered around the 30s
Note: The display could be constructed so that only five possible values (instead of ten) could fall in each stem. What would the stems look like? Would there be a difference in appearance?

16. Remember!
1. It is fairly typical of many variables to display a distribution that is concentrated (mounded) about a central value and then in some manner be dispersed in both directions. (Why?)
2. A display that indicates two “mounds” may really be two overlapping distributions
3. A back-to-back stem-and-leaf display makes it possible to compare two distributions graphically
4. A side-by-side dotplot is also useful for comparing two distributions

17. 2.2 ~ Frequency Distributions & Histograms
Stem-and-leaf plots often present adequate summaries, but they can get very big, very fast
Need other techniques for summarizing data
Frequency distributions and histograms are used to summarize large data sets

18. Frequency Distributions
Frequency Distribution: A listing, often expressed in chart form, that pairs each value of a variable with its frequency
Ungrouped Frequency Distribution: Each value of x in the distribution stands alone
Grouped Frequency Distribution: Group the values into a set of classes
1. A table that summarizes data by classes, or class intervals
2. In a typical grouped frequency distribution, there are usually 5-12 classes of equal width
3. The table may contain columns for class number, class interval, tally (if constructing by hand), frequency, relative frequency, cumulative relative frequency, and class midpoint
4. In an ungrouped frequency distribution each class consists of a single value

19. Frequency Distribution
Guidelines for constructing a frequency distribution:
1. All classes should be of the same width
2. Classes should be set up so that they do not overlap and so that each piece of data belongs to exactly one class
3. For problems in the text, 5-12 classes are most desirable. The square root of n is a reasonable guideline for the number of classes if n is less than 150.
4. Use a system that takes advantage of a number pattern, to guarantee accuracy
5. If possible, an even class width is often advantageous

20. Frequency Distributions
Procedure for constructing a frequency distribution:
1. Identify the high (H) and low (L) scores. Find the range. Range = H - L
2. Select a number of classes and a class width so that the product is a bit larger than the range
3. Pick a starting point a little smaller than L. Count from L by the width to obtain the class boundaries. Observations that fall on class boundaries are placed into the class interval to the right.

21. Example
Example: The hemoglobin test, a blood test given to diabetics during their periodic checkups, indicates the level of control of blood sugar during the past two to three months. The data in the table below was obtained for 40 different diabetics at a university clinic that treats diabetic patients:
1) Construct a grouped frequency distribution using the classes <4.7, <5.7, <6.7, etc.
2) Which class has the highest frequency?

22. Solutions 1)
Class Frequency Relative Cumulative Class
Boundaries f Frequency Rel. Frequency Midpoint, x
3.7 - <
4.7 - <
5.7 - <
6.7 - <
7.7 - <
8.7 - <
2) The class <6.7 has the highest frequency. The frequency is 16 and the relative frequency is 0.40

23. Histogram
Histogram: A bar graph representing a frequency distribution of a quantitative variable. A histogram is made up of the following components:
1. A title, which identifies the population of interest
2. A vertical scale, which identifies the frequencies in the various classes
3. A horizontal scale, which identifies the variable x. Values for the class boundaries or class midpoints may be labeled along the x-axis. Use whichever method of labeling the axis best presents the variable.
Notes:
The relative frequency is sometimes used on the vertical scale
It is possible to create a histogram based on class midpoints

24. Example
Example: Construct a histogram for the blood test results given in the previous example 9 . 2 8 7 6 5 4 1
Frequency
Blood Test
Solution:
The Hemoglobin Test

25. Example
Example: A recent survey of Roman Catholic nuns summarized their ages in the table below. Construct a histogram for this age data:
Age Frequency Class Midpoint
20 up to
30 up to
40 up to
50 up to
60 up to
70 up to
80 up to

26. Solution 8 5 7 6 4 3 2 1
Frequency
Age
Roman Catholic Nuns

27. Terms Used to Describe Histograms
Symmetrical: Both sides of the distribution are identical mirror images. There is a line of symmetry.
Uniform (Rectangular): Every value appears with equal frequency
Skewed: One tail is stretched out longer than the other. The direction of skewness is on the side of the longer tail. (Positively skewed vs. negatively skewed)
J-Shaped: There is no tail on the side of the class with the highest frequency
Bimodal: The two largest classes are separated by one or more classes. Often implies two populations are sampled.
Normal: A symmetrical distribution is mounded about the mean and becomes sparse at the extremes

28. Important Reminders
The mode is the value that occurs with greatest frequency (discussed in Section 2.3)
The modal class is the class with the greatest frequency
A bimodal distribution has two high-frequency classes separated by classes with lower frequencies
Graphical representations of data should include a descriptive, meaningful title and proper identification of the vertical and horizontal scales

29. Cumulative Frequency Distribution
Cumulative Frequency Distribution: A frequency distribution that pairs cumulative frequencies with values of the variable
The cumulative frequency for any given class is the sum of the frequency for that class and the frequencies of all classes of smaller values
The cumulative relative frequency for any given class is the sum of the relative frequency for that class and the relative frequencies of all classes of smaller values

30. Example
Example: A computer science aptitude test was given to students. The table below summarizes the data:
Class Relative Cumulative Cumulative
Boundaries Frequency Frequency Frequency Rel. Frequency
0 up to
4 up to
8 up to
12 up to
16 up to
20 up to
24 up to

31. Ogive
Ogive: A line graph of a cumulative frequency or cumulative relative frequency distribution. An ogive has the following components:
1. A title, which identifies the population or sample
2. A vertical scale, which identifies either the cumulative frequencies or the cumulative relative frequencies
3. A horizontal scale, which identifies the upper class boundaries. Until the upper boundary of a class has been reached, you cannot be sure you have accumulated all the data in the class. Therefore, the horizontal scale for an ogive is always based on the upper class boundaries.
Note: Every ogive starts on the left with a relative frequency of zero at the lower class boundary of the first class and ends on the right with a relative frequency of 100% at the upper class boundary of the last class.

32. Cumulative Relative Frequency Computer Science Aptitude Test
Example
Example: The graph below is an ogive using cumulative relative frequencies for the computer science aptitude data:
Cumulative Relative Frequency 4 8 12 16 20 24 28
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Test Score
Computer Science Aptitude Test

33. 2.3 ~ Measures of Central Tendency
Numerical values used to locate the middle of a set of data, or where the data is clustered
The term average is often associated with all measures of central tendency

34. Mean
Mean: The type of average with which you are probably most familiar. The mean is the sum of all the values divided by the total number of values, n: x n i = + å 1 2 ( )
The population mean, , (lowercase mu, Greek alphabet), is the mean of all x values for the entire population
Notes:
We usually cannot measure  but would like to estimate its value
A physical representation: the mean is the value that balances the weights on the number line

35. Example
Example: The following data represents the number of accidents in each of the last 8 years at a dangerous intersection. Find the mean number of accidents: 8, 9, 3, 5, 2, 6, 4, 5: x = + 1 8 9 3 5 2 6 4 25 ( ) .
Solution:
In the data above, change 6 to 26: x = + 1 8 9 3 5 2 26 4 7 75 ( ) .
Solution:
Note: The mean can be greatly influenced by outliers

36. Median
Median: The value of the data that occupies the middle position when the data are ranked in order according to size
Notes:
Denoted by “x tilde”:
The population median,  (uppercase mu, Greek alphabet), is the data value in the middle position of the entire population
To find the median:
1. Rank the data
2. Determine the depth of the median:
3. Determine the value of the median

37. Example Example: Find the median for the set of data:
{4, 8, 3, 8, 2, 9, 2, 11, 3}
Solution:
1. Rank the data: 2, 2, 3, 3, 4, 8, 8, 9, 11
2. Find the depth:
3. The median is the fifth number from either end in the ranked data:
Suppose the data set is {4, 8, 3, 8, 2, 9, 2, 11, 3, 15}:
1. Rank the data: 2, 2, 3, 3, 4, 8, 8, 9, 11, 15
2. Find the depth:
3. The median is halfway between the fifth and sixth observations:

38. Mode & Midrange
Mode: The mode is the value of x that occurs most frequently
Note: If two or more values in a sample are tied for the highest frequency (number of occurrences), there is no mode
Midrange: The number exactly midway between a lowest value data L and a highest value data H. It is found by averaging the low and the high values:

39. Example Example: Consider the data set {12.7, 27.1, 35.6, 44.2, 18.0}
Midrange = + L H 2 12 7 44 28 45 .
When rounding off an answer, a common rule-of-thumb is to keep one more decimal place in the answer than was present in the original data
To avoid round-off buildup, round off only the final answer, not intermediate steps
Notes:

40. 2.4 ~ Measures of Dispersion
Measures of central tendency alone cannot completely characterize a set of data. Two very different data sets may have similar measures of central tendency.
Measures of dispersion are used to describe the spread, or variability, of a distribution
Common measures of dispersion: range, variance, and standard deviation

41. Range
Range: The difference in value between the highest-valued (H) and the lowest-valued (L) pieces of data:
Other measures of dispersion are based on the following quantity
Deviation from the Mean: A deviation from the mean, , is the difference between the value of x and the mean

42. Example
Example: Consider the sample {12, 23, 17, 15, 18}. Find 1) the range and 2) each deviation from the mean.
Solutions: 1)
Data Deviation from Mean
_________________________ 12 23 17 15 18 2) -5 6 -2 1

43. Mean Absolute Deviation
Note: (Always!) å = - x ) (
Mean Absolute Deviation: The mean of the absolute values of the deviations from the mean: å - = x | 1
deviation
absolute
Mean n 8 . 2 5 14 ) 1 6 ( | = + - å x n
For the previous example:

44. Sample Variance & Standard Deviation
Sample Variance: The sample variance, s2, is the mean of the squared deviations, calculated using n - 1 as the divisor: s n x 2 1 = - å ( )
where n is the sample size ( ) SS x n = - å 2 1
Note: The numerator for the sample variance is called the sum of squares for x, denoted SS(x):
where
Standard Deviation: The standard deviation of a sample, s, is the positive square root of the variance:

45. Example
Example: Find the 1) variance and 2) standard deviation for the data {5, 7, 1, 3, 8}: x - ( ) 2
0.2
2.2
-3.8
-1.8
3.2
0.04
4.84
14.44
3.24
10.24 5 7 1 3 8 24
32.08
Sum:
Solutions: = + 4 .
First: 2 . 8 ) 32 ( 4 1 = s 1) 86 . 2 8 = s 2)

46. Notes The shortcut formula for the sample variance:
The unit of measure for the standard deviation is the same as the unit of measure for the data

47. 2.5 ~ Mean & Standard Deviation of Frequency Distribution
If the data is given in the form of a frequency distribution, we need to make a few changes to the formulas for the mean, variance, and standard deviation
Complete the extension table in order to find these summary statistics

48. To Calculate
In order to calculate the mean, variance, and standard deviation for data:
1. In an ungrouped frequency distribution, use the frequency of occurrence, f, of each observation
2. In a grouped frequency distribution, we use the frequency of occurrence associated with each class midpoint:

49. Example
Example: A survey of students in the first grade at a local school asked for the number of brothers and/or sisters for each child. The results are summarized in the table below. Find 1) the mean, 2) the variance, and 3) the standard deviation:
Solutions:
First: 17 23 5 2 46 20 10 1 4 62 93
Sum: 15 92 80 50
239 2) 1) 3)

50. TI-83 Calculations
When dealing with a grouped frequency distribution, use the following technique:
Input the class midpoints or data values into L1 and the frequencies into L2; then continue with
Highlight: L3
Enter: L3 = L1*L2
Highlight: L4
Enter: L4 = L1*L3
Highlight: L5(1) (first position in L5 column)

51. TI-83 Calculations Enter: L5(1) = sum(L2) (Σf) L5(2) = sum(L3) (Σxf)
To find sum use 2nd “List”>Math>5:sum(
L5(2) = sum(L3) (Σxf)
L5(3) = sum(L4) (Σx2f)
L5(4) = L5(2)/L5(1) to find the mean
L5(5) = (L5(3)-(L5(2))2/L5(1))/(L5(1)-1) to find the variance
L5(6) = 2nd √ (L5(5) to find the standard deviation
Let’s work problem as an example!

52. Problem 2.108
Find the mean and the variance for this grouped frequency distribution:
Class Boundaries f
2 –
6 –
10 –
14 –
18 –
Step 1: Enter the midpoints into L1
Step 2: Enter the frequencies into L2

53. Problem 2.108 Highlight L3 and enter L1 * L2
Highlight L5(1) and enter Sum(L2)

54. Problem 2.108 Highlight L5(2) and enter Sum(L3)
Highlight L5(4) and enter L5(2)/L5(1)
Highlight L5(5) and enter (L5(3)-(L5(2))2/L5(1))/(L5(1)-1)
Finally highlight L5(6) and enter 2nd √(L5(5))

55. Problem 2.73 Mean 9.77 9.80 -0.03 Median 9.65 9.78 -0.06 Maximum 13.65
Runs At Home
Runs Away
Difference
Mean
9.77
9.80
-0.03
Median
9.65
9.78
-0.06
Maximum
13.65
11.06
4.89
Minimum
7.64
8.67
-1.74
Midrange
10.65
9.87
1.58

56. Problem 2.73 cont.
Teams playing the Rockies at Coors Field generated the maximum number of runs scored (13.65). On the other hand, while playing their opponents away, the Rockies and their opponents, were able to generate only 8.76 runs, which ranked second from the bottom. Collectively, these two performances produced the greatest spread (4.89) by a considerable margin between runs scored at home and runs scored away by any stadium/team combination in the major leagues. This unusually large value inflates the midrange difference. It appears the playing conditions at Coors Field, therefore, are more responsible for producing the higher combined scores than the strength of either the Rockies’ or their opponents’ bats or any weakness of the pitching staffs.

57. Problem 2.75
∑ x need to be 500, therefore need any three numbers that total 330.
Need two numbers smaller than 70 and one larger
Need multiple 87’s
Need any two numbers that total 140 for the extreme values where one is 100 or larger
Need two numbers smaller than 70 and one larger than 70 so their total is 330
Need two numbers of 87 and a third number large enough so that the total of all five is 500.
Mean equal to 100 requires the five data to total 500 and the midrange of 70 requires the total of L and H to be 140; 40, __, 70, ___, 100; that is a sum of 210, meaning the other two data must total One of the last two numbers must be larger than 145, which would then become H and change the midrange. Impossible.
There must be two 87’s in order to have a mode, and there can only be two data larger than 70 in order for 70 to be the median. , 70, 87, 87, 100; Impossible

58. Problem 2.83 Range = H – L = 9 – 2 = 7 1st: find the mean: 6
s2 = ∑(x-xbar)2/(n-1) = 34/4 = 8.5
s = √s2 = √8.5 = 2.9 x
x – xbar
(x – xbar)2 2 -4 16 4 -2 7 1 8 9 3 ∑ 34

59. 2.6 ~ Measures of Position
Measures of position are used to describe the relative location of an observation
Quartiles and percentiles are two of the most popular measures of position
An additional measure of central tendency, the midquartile, is defined using quartiles
Quartiles are part of the 5-number summary

60. Ranked data, increasing order
Quartiles
Quartiles: Values of the variable that divide the ranked data into quarters; each set of data has three quartiles
1. The first quartile, Q1, is a number such that at most 25% of the data are smaller in value than Q1 and at most 75% are larger
2. The second quartile, Q2, is the median
3. The third quartile, Q3, is a number such that at most 75% of the data are smaller in value than Q3 and at most 25% are larger
Ranked data, increasing order

61. Percentiles
Percentiles: Values of the variable that divide a set of ranked data into 100 equal subsets; each set of data has 99 percentiles. The kth percentile, Pk, is a value such that at most k% of the data is smaller in value than Pk and at most (100 - k)% of the data is larger. ~ x Q P = 2 50
Notes:
The 1st quartile and the 25th percentile are the same: Q1 = P25
The median, the 2nd quartile, and the 50th percentile are all the same:

62. Finding Pk (and Quartiles)
Procedure for finding Pk (and quartiles):
1. Rank the n observations, lowest to highest
2. Compute A = (nk)/100
3. If A is an integer:
d(Pk) = A.5 (depth)
Pk is halfway between the value of the data in the Ath position and the value of the next data
If A is a fraction:
d(Pk) = B, the next larger integer
Pk is the value of the data in the Bth position

63. Example
Example: The following data represents the pH levels of a random sample of swimming pools in a California town. Find: 1) the first quartile, 2) the third quartile, and 3) the 37th percentile:
Solutions:
1) k = 25: (20) (25) / 100 = 5, depth = 5.5, Q1 = 6
2) k = 75: (20) (75) / 100 = 15, depth = 15.5, Q3 = 6.95
3) k = 37: (20) (37) / 100 = 7.4, depth = 8, P37 = 6.2

64. Midquartile
Midquartile: The numerical value midway between the first and third quartile:
Example: Find the midquartile for the 20 pH values in the previous example:
475 . 6 2 95 12 e
midquartil 3 1 = + Q
Note: The mean, median, midrange, and midquartile are all measures of central tendency. They are not necessarily equal. Can you think of an example when they would be the same value?

65. 5-Number Summary
5-Number Summary: The 5-number summary is composed of:
1. L, the smallest value in the data set
2. Q1, the first quartile (also P25)
3. , the median (also P50 and 2nd quartile)
4. Q3, the third quartile (also P75)
5. H, the largest value in the data set
Notes:
The 5-number summary indicates how much the data is spread out in each quarter
The interquartile range is the difference between the first and third quartiles. It is the range of the middle 50% of the data

66. Box-and-Whisker Display
Box-and-Whisker Display: A graphic representation of the 5-number summary:
The five numerical values (smallest, first quartile, median, third quartile, and largest) are located on a scale, either vertical or horizontal
The box is used to depict the middle half of the data that lies between the two quartiles
The whiskers are line segments used to depict the other half of the data
One line segment represents the quarter of the data that is smaller in value than the first quartile
The second line segment represents the quarter of the data that is larger in value that the third quartile

67. Example
Example: A random sample of students in a sixth grade class was selected. Their weights are given in the table below. Find the 5-number summary for this data and construct a boxplot:
Solution:

68. Boxplot for Weight Data
Weights from Sixth Grade Class 1 9 8 7 6
Weight

69. z-Score
z-Score: The position a particular value of x has relative to the mean, measured in standard deviations. The z-score is found by the formula:
Notes:
Typically, the calculated value of z is rounded to the nearest hundredth
The z-score measures the number of standard deviations above/below, or away from, the mean
z-scores typically range from to +3.00
z-scores may be used to make comparisons of raw scores

70. Example
Example: A certain data set has mean 35.6 and standard deviation Find the z-scores for 46 and 33:
Solutions:
46 is 1.46 standard deviations above the mean z x s = - 33 35 6 7 1 37 .
33 is 0.37 standard deviations below the mean.

71. 2.7 ~ Interpreting & Understanding Standard Deviation
Standard deviation is a measure of variability, or spread
Two rules for describing data rely on the standard deviation:
Empirical rule: applies to a variable that is normally distributed
Chebyshev’s theorem: applies to any distribution

72. Empirical Rule
Empirical Rule: If a variable is normally distributed, then:
1. Approximately 68% of the observations lie within 1 standard deviation of the mean
2. Approximately 95% of the observations lie within 2 standard deviations of the mean
3. Approximately 99.7% of the observations lie within 3 standard deviations of the mean
Notes:
The empirical rule is more informative than Chebyshev’s theorem since we know more about the distribution (normally distributed)
Also applies to populations
Can be used to determine if a distribution is normally distributed

73. Illustration of the Empirical Rule
68%
95%
99.7%

74. Example
Example: A random sample of plum tomatoes was selected from a local grocery store and their weights recorded. The mean weight was 6.5 ounces with a standard deviation of 0.4 ounces. If the weights are normally distributed:
1) What percentage of weights fall between 5.7 and 7.3?
2) What percentage of weights fall above 7.7?
Solutions: ( , ) .
(0. ), )) x s - + = 2 6 5 4 7 3
Approximat
ely 95% of
the weigh
ts fall be
tween 5.7
and 7.3 1) ( , ) .
(0. ), )) x s - + = 3 6 5 4 7
Approximat
ely 99.7%
of the wei
ghts fall
between 5.
3 and 7.7
ely 0.3% of
the weigh
ts fall out
side (5.3,
7.7)
ely (0.3 / 2)
=0.15% of t
he weights
fall abov
e 7.7

75. A Note about the Empirical Rule
Note: The empirical rule may be used to determine whether or not a set of data is approximately normally distributed
1. Find the mean and standard deviation for the data
2. Compute the actual proportion of data within 1, 2, and 3 standard deviations from the mean
3. Compare these actual proportions with those given by the empirical rule
4. If the proportions found are reasonably close to those of the empirical rule, then the data is approximately normally distributed

76. Chebyshev’s Theorem
Chebyshev’s Theorem: The proportion of any distribution that lies within k standard deviations of the mean is at least 1 - (1/k2), where k is any positive number larger than 1. This theorem applies to all distributions of data.
Illustration:

77. Important Reminders!
Chebyshev’s theorem is very conservative and holds for any distribution of data
Chebyshev’s theorem also applies to any population
The two most common values used to describe a distribution of data are k = 2, 3
The table below lists some values for k and 1 - (1/k2):

78. Example
Example: At the close of trading, a random sample of 35 technology stocks was selected. The mean selling price was and the standard deviation was Use Chebyshev’s theorem (with k = 2, 3) to describe the distribution.
Solutions:
Using k=2: At least 75% of the observations lie within 2 standard deviations of the mean:
Using k=3: At least 89% of the observations lie within 3 standard deviations of the mean:

79. 2.8 ~ The Art of Statistical Deception
Good Arithmetic, Bad Statistics
Misleading Graphs
Insufficient Information

80. Good Arithmetic, Bad Statistics
The mean can be greatly influenced by outliers
Example: The mean salary for all NBA players is $15.5 million
Misleading graphs:
1. The frequency scale should start at zero to present a complete picture. Graphs that do not start at zero are used to save space.
2. Graphs that start at zero emphasize the size of the numbers involved
3. Graphs that are chopped off emphasize variation

81. Flight Cancellations 2 1 9 98 6 3 5
Number of
Cancellations
Year

82. Flight Cancellations 3 5 4 2 1 9 8 7 98 6
Year
Number of
Cancellations

83. Insufficient Information
Example: An admissions officer from a state school explains that the average tuition at a nearby private university is $13,000 and only $4500 at his school. This makes the state school look more attractive.
If most students pay the full tuition, then the state school appears to be a better choice
However, if most students at the private university receive substantial financial aid, then the actual tuition cost could be much lower!