1. Chapter 2 Descriptive Statistics
2-1 Overview
2-2 Summarizing Data
2-3 Pictures of Data
2-4 Measures of Central Tendency
2-5 Measures of Variation
2-6 Measures of Position
2-7 Exploratory Data Analysis
Review and Projects

2. Overview 2-1 Descriptive Statistics Inferential Statistics
summarizes or describes the important characteristics of a known set of population data
Inferential Statistics
uses sample data to make inferences about a population

3. Important Characteristics
of Data
1. Nature or shape of the distribution, such as bell-shaped, uniform, or skewed
2. Representative score, such as an average
3. Measure of scattering or variation

4. Summarizing Data With Frequency Tables
2-2
Summarizing Data With Frequency Tables
Frequency Table
lists categories (or classes) of scores, along with counts (or frequencies) of the number of scores that fall into each category

5. Axial Loads of 0.0109 in. Cans Table 2-1 270 278 250 290 274 242 269
257
272
265
263
234
273
277
294
279
268
230
262
273
201
275
260
286
272
284
282
278
268
263
285
289
208
292
279
276
242
258
264
281
262
278
265
241
267
295
283
209
276
273
263
218
271
289
223
217
225
292
270
204
265
271
273
283
275
276
282
270
256
268
259
272
269
251
208
290
220
277
293
254
223
263
274
262
200
272
268
206
280
287
257
284
279
252
215
281
291
276
285
297
290
228
274
277
286
251
278
289
269
267
276
206
284
268
291
293
280
282
230
275
236
295
289
283
261
262
252
277
204
286
270
278
272
281
288
248
266
256
292

6. Frequency Table of Axial Loads of Aluminum Cans 9 3 5 4 14 32 52 38

7. Frequency Table Definitions
Class: An interval.
Lower Class Limit: The left endpoint of a class.
Upper Class Limit: The upper endpoint of a class.
Class Mark: The midpoint of the class.
Class width: the difference between the two consecutive lower class limits.

8. Definition values for the example
Score
Frequency 9 3 5 4 14 32 52 38
Table 2-2
Lower Class Limits: 200, 210, …
Upper class limits: 209,219 …
Class Marks: 204.5=( )/2,, 214.5, …
Class width: =10.

9. Determine the Definition Values for this Frequency Table
Classes
Lower Class Limits
Upper Class Limits
Class Marks
Class Width
Quiz
Scores
Frequency 2 5 8 11 7
0 - 4
5 - 9

10. class width = round up of
Constructing A Frequency Table
1. Decide on the number of classes.
2. Determine the class width by dividing the range by the number of classes (range = highest score – lowest score) and round up.
range
class width = round up of
number of classes
3. Select for the first lower limit either the lowest score or a convenient value slightly less than the lowest score.
4. Add the class width to the starting point to get the second lower class limit.
5. List the lower class limits in a vertical column and enter the upper class limits.
6. Represent each score by a tally mark in the appropriate class.
Total tally marks to find the total frequency for each class.

11. Guidelines For Frequency Tables
1. Classes should be mutually exclusive.
2. Include all classes, even if the frequency is zero.
3. Try to use the same width for all classes.
4. Select convenient numbers for class limits.
5. Use between 5 and 20 classes.
6. The sum of the class frequencies must equal the number of original data values.

12. Relative Frequency Table
class frequency
sum of all frequencies

13. Relative Frequency Table
Score
Frequency 9 3 5 4 14 32 52 38
Table 2-2
Axial
Load
Relative
0.051
0.017
0.029
0.023
0.080
0.183
0.297
0.217
0.080-
Table 2-3 9
= .051
175 3
= .017
175 5
= .029
175

14. Cumulative Frequency Table
Axial
Load
Cumulative
Frequency
Score
Frequency 9 3 5 4 14 32 52 38
Less than 210
Less than 220
Less than 230
Less than 240
Less than 250
Less than 260
Less than 270
Less than 280
Less than 290
Less than 300 9 12 17 21 25 39 71
123
161
175
Cumulative
Frequencies

15. Frequency Tables Table 2-2 Table 2-3 Table 2-4 Axial Load Relative
Cumulative
Frequency
Score
Frequency
Less than 210
Less than 220
Less than 230
Less than 240
Less than 250
Less than 260
Less than 270
Less than 280
Less than 290
Less than 300 9 12 17 21 25 39 71
123
161
175 9 3 5 4 14 32 52 38
0.051
0.017
0.029
0.023
0.080
0.183
0.297
0.217
0.08-

16. Mean as a Balance Point
Mean
FIGURE 2-7

17. Notation µ is pronounced ‘mu’ and denotes the mean of all values
S denotes the summation of a set of values
x is the variable usually used to represent the individual data values
n represents the number of data values in a sample
N represents the number of data values in a population
x is pronounced ‘x-bar’ and denotes the mean of a set of
sample values
µ is pronounced ‘mu’ and denotes the mean of all values
in a population

18. Calculators can calculate the mean of data
Definitions
Mean
the value obtained by adding the scores and dividing the total by the number of scores
S x
Sample
x = n
S x
µ =
Population N
Calculators can calculate the mean of data

19. Definitions Median often denoted by x (pronounced ‘x-tilde’)
the middle value when scores are arranged in (ascending or descending) order ~
often denoted by x (pronounced ‘x-tilde’)
is not affected by an extreme value

20. no exact middle -- shared by two numbers
(in order)
exact middle MEDIAN is 4
no exact middle -- shared by two numbers
MEDIAN is 4.5
4 + 5
= 4.5 2

21. Definitions Mode Bimodal Multimodal No Mode
the score that occurs most frequently
Bimodal
Multimodal
No Mode
the only measure of central tendency that can be used with nominal data

22. Examples a b c
Mode is 5
Bimodal
No Mode

23. Examples a b c
Mode is 5
Bimodal
No Mode d e
Mode is 3
No Mode

24. highest score + lowest score
Definitions
Midrange
the value halfway between the highest and lowest scores
highest score + lowest score
Midrange = 2

25. measures of central tendency
Round-off rule for
measures of central tendency
Carry one more decimal place than is present in the orignal set of data

26. An Example of Skewness Symmetric Dataset 1: 3, 4, 4, 5, 5, 5, 6, 6, 7
Mean = 5, Median = 5
Dataset 2: 3, 4, 4, 5, 5, 5, 7, 7 ,9.
Mean=5.444, Median = 5.
Skewed right
Dataset 3: 2, 3, 3, 5, 5, 5, 6, 6, 7.
Mean = 4.667, Median = 5.
Skewed left

27. Skewness SYMMETRIC SKEWED LEFT SKEWED RIGHT (negatively) (positively)
Figure
2-8 (b)
Mode = Mean = Median
SYMMETRIC
Mean
Mode
Mode
Mean
Median
Median
Figure
2-8 (a)
SKEWED LEFT
(negatively)
SKEWED RIGHT
(positively)
Figure
2-8 (c)

28. Best Measure
of Central Tendency
Table 2-6
Advantages - Disadvantages

29. Mean from a Frequency Table use class mark of classes for variable x
S (f • x)
x = Formula 2-2 S f
x = class mark
f = frequency
S f = n

30. Mean of this frequency table =14.4
Quiz Scores
Frequency
Class Marks
0 - 4
5 - 9 2 5 8 11 7 2 7 12 17 22
Mean of this frequency table =14.4

31. Measure of Variation
Range
score
highest
lowest
score

32. (average deviation from the mean)
Measure of Variation
Standard Deviation
a measure of variation of the scores about the mean
(average deviation from the mean)

33. Sample Standard Deviation Formula
S (x – x)2
S =
n – 1
Formula 2 -4
calculators can calculate sample standard deviation of data

34. Find the standard deviation of the sample data:
2, 3, 4, 5, 5, 5. S2 = 8/5=1.6, S=1.26.
Use the shortcut formula to find the standard deviations of the above data, and the waiting times at the two banks.
1) S x2 =104,
2) Jefferson Valley Bank: S x2 =513.27, S x =71.5, s=0.48.
3) Bank of Providence: S x2 =541.09, S x =71.5, s=1.82.

35. Population Standard Deviation
S (x – µ) 2
s = N
calculators can calculate the population standard deviation of data

36. for Standard Deviation
Symbols
for Standard Deviation
Sample
Population s Sx
xsn–1 s
s x
xsn
Textbook
Book
Some graphics
calculators
Some graphics
calculators
Some
nongraphics
calculators
Some
nongraphics
calculators

37. standard deviation squared
Measure of Variation
Variance
standard deviation squared s } 2
use square key
on calculator
Notation 2

38. Variance S (x – x)2 n – 1 S (x – µ)2 s2 = N s2 = Sample Variance
Population
Variance N

39. Round-off Rule for measures of variation
Carry one more decimal place than was present in the original data

40. Standard Deviation Shortcut Formula
n (S x2) – (S x)2
s =
n (n – 1)
Formula 2 - 6

41. Standard deviation gets larger as spread of data increases.
FIGURE 2-10
Same Means (x = 4)
Different Standard Deviations
s = 0 1 2 3 4 5 6 7
s = 3.0
s = 0.8
s = 1.0
Frequency
Standard deviation gets larger as spread of data increases.

42. (applies to bell shaped distributions)
FIGURE 2-10
The Empirical Rule
(applies to bell shaped distributions)
68% within
1 standard deviation
0.340
0.340
x – s x
x + s

43. (applies to bell shaped distributions)
FIGURE 2-10
The Empirical Rule
(applies to bell shaped distributions)
95% within
2 standard deviations
68% within
1 standard deviation
0.340
0.340
0.135
0.135
x – 2s
x – s x
x + s
x + 2s

44. The Empirical Rule x – 3s x – 2s x – s x x + s x + 2s x + 3s
FIGURE 2-10
The Empirical Rule
(applies to bell shaped distributions)
99.7% of data are within 3 standard deviations of the mean
95% within
2 standard deviations
68% within
1 standard deviation
0.340
0.340
0.024
0.024
0.001
0.001
0.135
0.135
x – 3s
x – 2s
x – s x
x + s
x + 2s
x + 3s

45. Range Rule of Thumb s » Range » 4s Range 4 (maximum) (minimum) or
x s
(maximum)
(minimum)
x – 2s x
Range » 4s or
s »
Range 4

46. Chebyshev’s Theorem applies to distributions of any shape
the proportion (or fraction) of any set of data lying within k standard deviations of the mean is always at least 1 – 1/k2, where k is any positive number greater than 1.

47. Measures of Variation Summary
For typical data sets, it is unusual for a score to differ from the mean by more than 2 or 3 standard deviations.

48. An application of measure of variation
There are two brands, A, B or car tires. Both have a mean life time of 60,000 miles, but brand A has a standard deviation on lifetime of 1000 miles and Brand B has a standard deviation on lifetime of 3000 miles. Which brand would you prefer?

49. divides ranked scores into four equal parts
Quartiles
Q1, Q2, Q3
divides ranked scores into four equal parts
25%
25%
25%
25% Q1 Q2 Q3

50. Percentiles
99 Percentiles

51. Sorted Axial Loads of 175 Aluminum Cans
Finding the Percentile of a Given Score
number of scores less than x
Percentile of score x = • 100
total number of scores
Sorted Axial Loads of 175 Aluminum Cans
[1]
[16]
[31]
[46]
[61]
[76]
[91]
[106]
[121]
[136]
[151]
[166]

52. Finding the Value of the kth Percentile
Start
Rank the data.
(Arrange the data in
order of lowest to
highest.)
Finding the Value of the kth Percentile
Compute
L = n where
n = number of scores
k = percentile in question ) ( k
100
The value of the kth percentile
is midway between the Lth score
and the highest score in the
original set of data. Find Pk by
adding the L th score and the
next higher score and dividing the
total by 2. Is
L a whole
number ?
Yes No
Change L by rounding
it up to the next
larger whole number.
The value of Pk is the
Lth score, counting from the lowest

53. Sorted Axial Loads of 175 Aluminum Cans
[1]
[16]
[31]
[46]
[61]
[76]
[91]
[106]
[121]
[136]
[151]
[166]
The 10th percentile: L=175*10/100=17.5, round up to 18. So the 10th
percentile is the 18th one in the sorted data, i.e., 230.
The 25th percentile: L=175*25/100=43.52, rounded up to 44. The 25th
percentile is the 44th one in the sorted data, I.ei. 262.

54. Interquartile Range: Q3 – Q1 Semi-interquartile Range: Midquartile: 2
Q1 + Q3 2

55. Exploratory Data Analysis
Used to explore data at a preliminary level
Few or no assumptions are made about the data
Tends to evolve relatively simple calculations and graphs

56. Exploratory Data Analysis Traditional Statistics
Used to explore data at a preliminary level
Few or no assumptions are made about the data
Tends to evolve relatively simple calculations and graphs
Traditional Statistics
Used to confirm final conclusions about data
Typically requires some very important assumptions about the data
Calculations are often complex, and graphs are often unnecessary

57. Boxplots Box-and-Whisker Diagram
5 - number summary
Minimum
first quartile Q1
Median
third quartile Q3
Maximum

58. Boxplots Box-and-Whisker Diagram60
68.5 78 52 90
Figure Boxplot of Pulse Rates (Beats per minute) of Smokers

59. Figure Boxplots
Normal
Uniform
Skewed

60. Values that are very far away from most of the data
Outliers
Values that are very far away from most of the data

61. Class Survey Data
Boxplots for the heights of those who never broke a bone and those who did

62. When comparing two or more boxplots, it is necessary to use the same scale.40 50 60 70 80 90
100
PULSE 1 2
(yes) SMOKE (No)