1. Descriptive Statistics2
Descriptive Statistics
Elementary Statistics
Larson Farber

2. Frequency Distributions and Their Graphs
Section 2.1
Frequency Distributions and Their Graphs

3. Frequency Distributions
Minutes Spent on the Phone
This data will be used for several examples. You might want to duplicate it for use with slides that follow.
Make a frequency distribution table with five classes.
Minimum value =
Maximum value = 67
Key values:
125

4. Steps to Construct a Frequency Distribution
1. Choose the number of classes
Should be between 5 and 15. (For this problem use 5)
2. Calculate the Class Width
Find the range = maximum value – minimum. Then divide
this by the number of classes. Finally, round up to a
convenient number. ( ) / 5 = Round up to 12.
3. Determine Class Limits
The lower class limit is the lowest data value that belongs in
a class and the upper class limit is the highest. Use the
minimum value as the lower class limit in the first class. (67)
4. Mark a tally | in appropriate class for each data value.
After all data values are tallied, count the tallies in each
class for the class frequencies.

5. Construct a Frequency Distribution
Minimum = 67, Maximum = 125
Number of classes = 5
Class width = 12 3 5 8 9
Class Limits Tally 78 90
102
114
126 67 79 91
103
115
Do all lower class limits first.

6. Frequency Histogram Boundaries Class 3 66.5 - 78.5 67 - 78 78.5 - 90.5
Class 3 5 8 9
Time on Phone 9 9 8 8 7 6 5 5 5
A histogram is a bar graph for which the bars touch. To form boundaries, find the distance between consecutive classes. Add half that distance to the lower limits and half to the upper limits. In this case the distance is 1 unit so add .5 to all upper limits and subtract .5 from all lower ones. The data must be quantitative. This histogram is labeled at the class boundaries. Explain that midpoints could have been labeled instead. 4 3 3 2 1 6 6 . 5 7 8 . 5 9 . 5 1 2 . 5 1 1 4 . 5 1 2 6 . 5
minutes

7. Frequency Polygon Time on Phone minutes Class 3 67 - 78 79 - 90 5 3 5 8 9
Time on Phone 9 9 8 8 7 6 5 5 5 4 3 3 2 1
The frequency polygon is labeled at midpoints. Explain that midpoints could have been labeled instead.
72.5
84.5
96.5
108.5
120.5
minutes
Mark the midpoint at the top of each bar. Connect consecutive midpoints. Extend the frequency polygon to the axis.

8. Other Information Midpoint: (lower limit + upper limit) / 2
Relative frequency: class frequency/total frequency
Cumulative frequency: number of values in that class or in lower
Class
Midpoint
Relative
Frequency
Cumulative
Frequency
( )/2
3/30 3 5 8 9
72.5
84.5
96.5
108.5
120.5
0.10
0.17
0.27
0.30 3 8 16 25 30
The first two columns reflect the work done in previous slides. Once the first midpoint is calculated, the others can be found by adding the class width to the previous midpoint. Notice the last entry in the cumulative frequency column is equal to the total frequency.

9. Relative Frequency Histogram
Time on Phone
Relative frequency
A relative frequency histogram will have the same shape as a frequency histogram.
minutes
Relative frequency on vertical scale

10. Ogive An ogive reports the number of values in the data set that
are less than or equal to the given value, x.
Minutes on Phone 3 8 16 25 30
66.5
78.5
90.5
102.5
114.5
126.5 10 20 30
Cumulative Frequency
Label each boundary on the horizontal axis. Start with 0 for the lower boundary of the first class. Then mark points corresponding to cumulative frequencies at the upper boundaries. Connect with line segments.
minutes

11. More Graphs and Displays
Section 2.2
More Graphs and Displays

12. Stem-and-Leaf Plot
Lowest value is 67 and highest value is 125, so list stems from 6 to 12.
Stem
Leaf
6 |
7 |
8 |
9 |
10 |
11 |
12 | 6 2
Divide each data value into a stem and a leaf. The leaf is the rightmost significant digit. The stem consists of the digits to the left. The data shown represent the first line of the ‘minutes on phone’ data used earlier. The complete stem and leaf will be shown on the next slide. 2 8 3
To see complete display, go to next slide. 4

13. Stem-and-Leaf Plot Key: 6 | 7 means 67 6 | 7 7 | 1 8 8 | 2 5 6 7 7
6 | 7
7 | 1 8
8 |
9 |
10 |
11 | 2 6 8
12 | 2 4 5
Key: 6 | 7 means 67
Stress the importance of using a key to explain the plot. 6|7 could mean 6700 or .067 for a different problem. A stem and leaf should not be used with data when values are very different such as 3, 34,900, 24 etc. The stem-and leaf has the advantage over a histogram of retaining the original values.

14. Stem-and-Leaf with two lines per stem
Key: 6 | 7 means 67
6 | 7
7 | 1
7 | 8
8 | 2
8 |
9 | 2
9 |
10 |
10 |
11 | 2
11 | 6 8
12 | 2 4
12 | 5
1st line digits
2nd line digits
With two lines per stem the data is more finely “chopped”. Class width is 5 times the leaf unit. All stems except possibly the first and last must have two lines even if one is blank. For this data set, the first line for the stem 6 can be blank because there are no data values from 60 to 64.
1st line digits
2nd line digits

15. Dot Plot
Phone 66 76 86 96
106
116
126
minutes
Dot plots also allow you to retain original values.

16. NASA budget (billions of $) divided among 3 categories.
Pie Chart
Used to describe parts of a whole
Central Angle for each segment
NASA budget (billions of $) divided among 3 categories.
Pie charts help visualize the relative proportion of each category. Find the relative frequency for each category and multiply it by 360 degrees to find the central angle.
Billions of $
Human Space Flight
Technology
Mission Support
Construct a pie chart for the data.

17. Pie Chart Human Space Flight 5.7 143 Technology 5.9 149
Billions of $
Degrees
Human Space Flight
5.7
143
Technology
5.9
149
Mission Support
2.7 68
14.3
360
Total
Mission
Support
19%
Human
Space Flight
40%
NASA Budget
(Billions of $)
Technology
41%

18. Scatter Plot Absences Grade x 8 2 5 12 15 9 6 y 78 92 90 58 43 74 81
Final
grade
(y) 40 45 50 55 60 65 70 75 80 85 90 95 2 4 6 8 10 12 14 16
Absences (x)

19. Measures of Central Tendency
Section 2.3
Measures of Central Tendency

20. Measures of Central Tendency
Mean: The sum of all data values divided by the number of values
For a population: For a sample:
Median: The point at which an equal number of values fall above and fall below
There are other measures of central tendency (ex. midrange), but these three are the most commonly used. Be sure to discuss the advantages and disadvantages for each. Discuss the sigma notation.
Mode: The value with the highest frequency

21. Calculate the mean, the median, and the mode
An instructor recorded the average number of absences for his students in one semester. For a random sample the data are:
Calculate the mean, the median, and the mode
Mean:
Median: Sort data in order
As students which measure is the most representative of the center of the data.
The middle value is 3, so the median is 3.
Mode: The mode is 2 since it occurs the most times.

22. Calculate the mean, the median, and the mode.
Suppose the student with 40 absences is dropped from the course. Calculate the mean, median and mode of the remaining values. Compare the effect of the change to each type of average.
Calculate the mean, the median, and the mode.
Mean:
Median: Sort data in order.
Ask students which measure was changed the most by eliminating the extreme value (outlier).
The middle values are 2 and 3, so the median is 2.5.
Mode: The mode is 2 since it occurs the most times.

23. Shapes of Distributions
Symmetric
Uniform
Mean = Median
Skewed right
Skewed left
Both curves at the top are symmetric. Note that the fulcrum is placed at the mean of each distribution. The direction of skew is with the tail. The mean is always in the direction of the skew.
Mean > Median
Mean < Median

24. Section 2.4
Measures of Variation

25. Two Data Sets Stock A Stock B
Closing prices for two stocks were recorded on ten successive Fridays. Calculate the mean, median and mode for each.
Stock A
Stock B
Use this example to review the measures of central tendency. Both sets of data have the same mean, the same median and the same mode. Students clearly see that the data sets are vastly different though. A good lead in for measures of variation.
Mean = 61.5
Median = 62
Mode = 67
Mean = 61.5
Median = 62
Mode = 67

26. Measures of Variation Range = Maximum value – Minimum value
Range for A = 67 – 56 = $11
Range for B = 90 – 33 = $57
The range is easy to compute but only uses two numbers from a data set.
Explain that if only one number changes, the range can be vastly changed. If a $56 stock dropped to $12 the range would change. Emphasize the need for different symbols for population parameters and sample statistics.

27. Measures of Variation
To learn to calculate measures of variation that use each and every value in the data set, you first want to know about deviations.
The deviation for each value x is the difference between the value of x and the mean of the data set.
Explain that if only one number changes, the range can be vastly changed. If a $56 stock dropped to $12 the range would change. Emphasize the need for different symbols for population parameters and sample statistics.
In a population, the deviation for each value x is:
In a sample, the deviation for each value x is:

28. Deviations Stock A Deviation 56 57 58 61 63 67 – 5.5 – 4.5 – 3.5 – 0.5
1.5
5.5
56 – 61.5
56 – 61.5
57 – 61.5
58 – 61.5
Make sure students know that the sum of the deviations from the mean is always 0.
The sum of the deviations is always zero.

29. Population Variance Population Variance: The sum of the squares of the
deviations, divided by N. x 56 57 58 61 63 67
– 5.5
– 4.5
– 3.5
– 0.5
1.5
5.5
30.25
20.25
12.25
0.25
2.25
188.50
Each deviation is squared to eliminate the negative sign. Students should know how to calculate these formulas for small data sets. For larger data sets they will use calculators or computer software programs.
Sum of squares

30. Population Standard Deviation
Population Standard Deviation: The square root of the population variance.
The variance is expressed in “square units” which are meaningless. Using a standard deviation returns the data to its original units.
The population standard deviation is $4.34.

31. Sample Variance and Standard Deviation
To calculate a sample variance divide the sum of squares by n – 1.
The sample standard deviation, s, is found by taking the square root of the sample variance.
Samples only contain a portion of the population. This portion is likely to not contain extreme values. Dividing by n-1 gives a higher value than dividing by n and so increases the standard deviation.

32. Summary Range = Maximum value – Minimum value Population Variance
Population Standard Deviation
Sample Variance
A summary of the formulas
Sample Standard Deviation

33. Empirical Rule ( %)
Data with symmetric bell-shaped distribution have the following characteristics.
13.5%
13.5%
2.35%
2.35% –4 –3 –2 –1 1 2 3 4
This type of distribution will be studied in much more detail in later chapters.
About 68% of the data lies within 1 standard deviation of the mean
About 95% of the data lies within 2 standard deviations of the mean
About 99.7% of the data lies within 3 standard deviations of the mean

34. Using the Empirical Rule
The mean value of homes on a street is $125 thousand with a standard deviation of $5 thousand. The data set has a bell shaped distribution. Estimate the percent of homes between $120 and $135 thousand.
125
130
135
120
140
145
115
110
105
$120 thousand is 1 standard deviation below
the mean and $135 thousand is 2 standard
deviations above the mean.
68% % = 81.5%
So, 81.5% have a value between $120 and $135 thousand.

35. Chebychev’s Theorem
For any distribution regardless of shape the portion of data lying within k standard deviations (k > 1) of the mean is at least 1 – 1/k2.
For k = 2, at least 1 – 1/4 = 3/4 or 75% of the data lie
within 2 standard deviation of the mean.
The empirical rule is valid for symmetric, mound -shaped distributions. Chebychev’s theorem applies to all distributions. Emphasize the phrase “at least” in the statement of the theorem.
For k = 3, at least 1 – 1/9 = 8/9 = 88.9% of the data lie within 3 standard deviation of the mean.

36. Chebychev’s Theorem
The mean time in a women’s 400-meter dash is 52.4 seconds with a standard deviation of 2.2 sec. Apply Chebychev’s theorem for k = 2.
Mark a number line in
standard deviation units.
2 standard deviations A
Apply Chebychev’s theorem for k = 3. At least 1-19 = 8/9 or the data values will fall within 3 standard deviation of the mean no matter what the distribution. At least 8/9 of the times in this race will fall between 45.8 and 59 sec.
45.8 48
50.2
52.4
54.6
56.8 59
At least 75% of the women’s 400-meter dash times will fall between 48 and 56.8 seconds.

37. Section 2.5
Measures of Position

38. Quartiles
3 quartiles Q1, Q2 and Q3 divide the data into 4 equal parts.
Q2 is the same as the median.
Q1 is the median of the data below Q2.
Q3 is the median of the data above Q2.
You are managing a store. The average sale for each of 27 randomly selected days in the last year is given. Find Q1, Q2, and Q3.

39. Finding Quartiles The data in ranked order (n = 27) are:
Median rank (27 + 1)/2 = 14. The median = Q2 = 42.
There are 13 values below the median.
Q1 rank= 7. Q1 is 30.
Q3 is rank 7 counting from the last value. Q3 is 45.
Calculators and computers might use a slightly different method. Values may differ, but only in small amounts.
The Interquartile Range is Q3 – Q1 = 45 – 30 = 15.

40. Box and Whisker Plot
A box and whisker plot uses 5 key values to describe a set of data. Q1, Q2 and Q3, the minimum value and the maximum value. Q1
Q2 = the median Q3
Minimum value
Maximum value 30 42 45 17 55 42 45 30 17
The Interquartile range is the difference between the third quartile and the first. It gives the range of the middle 50% of the data. 55 15 25 35 45 55
Interquartile Range = 45 – 30 = 15

41. Percentiles
Percentiles divide the data into 100 parts. There are 99 percentiles: P1, P2, P3…P99.
P50 = Q2 = the median
P25 = Q1
P75 = Q3
There are several different ways to calculate percentiles, so the example is only meant to serve as an approximation. Later in the course students will use cumulative areas to calculate probabilities. Percentiles are a good tie in.
A 63rd percentile score indicates that score is greater than or equal to 63% of the scores and less than or equal to 37% of the scores.

42. Percentiles Cumulative distributions can be used to find percentiles.
There are several different ways to calculate percentiles, so the example is only meant to serve as an approximation. Later in the course students will use cumulative areas to calculate probabilities. Percentiles are a good tie in.
114.5 falls on or above 25 of the 30 values.
25/30 =
So you can approximate 114 = P83.

43. Standard Scores
The standard score or z-score, represents the number of standard deviations that a data value, x, falls from the mean.
The test scores for a civil service exam have a mean of 152 and standard deviation of 7. Find the standard z-score for a person with a score of:
(a) (b) (c) 152
This person scores 1.29 standard deviations above the mean.
this person scores .57 standard deviations below the mean.
The score for this person is equal to the mean.

44. Calculations of z-Scores
A value of x = 161 is 1.29 standard deviations above the mean.
(b)
A value of x = 148 is 0.57 standard deviations below the mean.
(c)
A value of x = 152 is equal to the mean.