Download Chapter 2

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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
1
2-1
Overview
 Descriptive 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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
2
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
3
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
4
Table 2-1
Axial Loads of 0.0109 in. Cans
270
278
250
278
290
274
242
269
257
272
265
263
234
270
273
270
277
294
279
268
230
268
278
268
262
273
201
275
260
286
272
284
282
278
268
263
273
282
285
289
268
208
292
275
279
276
242
285
273
268
258
264
281
262
278
265
241
267
295
283
281
209
276
273
263
218
271
289
223
217
225
283
292
270
262
204
265
271
273
283
275
276
282
270
256
268
259
272
269
270
251
208
290
220
259
282
277
282
256
293
254
223
263
274
262
263
200
272
268
206
280
287
257
284
279
252
280
215
281
291
276
285
287
297
290
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
228
274
277
286
277
251
278
277
286
277
289
269
267
276
206
284
269
284
268
291
289
293
277
280
274
282
230
275
236
295
289
283
261
262
252
283
277
204
286
270
278
270
283
272
281
288
248
266
256
292
5
Table 2-2
Frequency Table of Axial
Loads of Aluminum Cans
Axial Load
Frequency
200 - 209
9
210 - 219
3
220 - 229
5
230 - 239
4
240 - 249
4
250 - 259
14
260 - 269
32
270 - 279
52
280 - 289
38
290 - 299
14
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
6
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.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
7
Definition values for the
example
Table 2-2
Score
Frequency
200 - 209
9
210 - 219
Lower Class Limits: 200, 210, …
3
220 - 229
Upper class limits: 209,219 …
5
230 - 239
4
240 - 249
4
250 - 259
14
260 - 269
32
270 - 279
52
280 - 289
38
290 - 299
14
Class Marks: 204.5=(200+209)/2,,
214.5, …
Class width: 210-200=10.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
8
Determine the Definition Values
for this Frequency Table
 Classes
Quiz
Scores
Frequency
0-4
2
5-9
5
10 - 14
8
15 - 19
11
20 - 24
7
 Lower Class Limits
 Upper Class Limits
 Class Marks
 Class Width
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
9
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.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
10
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.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
11
Relative Frequency Table
relative frequency =
class frequency
sum of all frequencies
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
12
Relative Frequency Table
Table 2-2
Score
Table 2-3
Frequency
Axial
Load
Relative
Frequency
200 - 209
9
200 - 209
0.051
210 - 219
3
210 - 219
0.017
220 - 229
5
220 - 229
0.029
230 - 239
4
230 - 239
0.023
240 - 249
4
240 - 249
0.023
250 - 259
14
250 - 259
0.080
260 - 269
32
260 - 269
0.183
270 - 279
52
270 - 279
0.297
280 - 289
38
280 - 289
0.217
290 - 299
14
290 - 299
0.080
9
= .051
175
3
= .017
175
5 = .029
175
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
13
Cumulative Frequency Table
Table 2-2
Score
Table 2-4
Frequency
Axial
Load
Cumulative
Frequency
200 - 209
9
Less than 210
9
210 - 219
3
Less than 220
12
220 - 229
5
Less than 230
17
230 - 239
4
Less than 240
21
240 - 249
4
Less than 250
25
250 - 259
14
Less than 260
39
260 - 269
32
Less than 270
71
270 - 279
52
Less than 280
123
280 - 289
38
Less than 290
161
290 - 299
14
Less than 300
175
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
Cumulative
Frequencies
14
Frequency Tables
Table 2-3
Table 2-2
Score
Frequency
Axial
Load
Relative
Frequency
200 - 209
9
200 - 209
0.051
210 - 219
3
210 - 219
0.017
220 - 229
5
220 - 229
0.029
230 - 239
4
230 - 239
0.023
240 - 249
4
240 - 249
0.023
250 - 259
14
250 - 259
0.080
260 - 269
32
260 - 269
0.183
270 - 279
52
270 - 279
0.297
280 - 289
38
280 - 289
0.217
290 - 299
14
290 - 299
0.08-
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
Table 2-4
Axial
Load
Cumulative
Frequency
Less than 210
9
Less than 220
12
Less than 230
17
Less than 240
21
Less than 250
25
Less than 260
39
Less than 270
71
Less than 280
123
Less than 290
161
Less than 300
175
15
Mean as a Balance Point
Mean
FIGURE 2-7
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
16
Notation
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
17
Definitions
 Mean
the value obtained by adding the scores and
dividing the total by the number of scores
Sample
Population
x =
Sx
n
Sx
µ =
N
Calculators can calculate the mean of data
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
18
Definitions
 Median
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
19
•
•
5
5
5
3
1
5
1
4
3
5
2
1 1 2
(in order)
3
3
4
5
5
5
5
5
exact middle
•
1
1
3
3
4
MEDIAN is 4
5
5
5
5
5
no exact middle -- shared by two numbers
4+5
= 4.5
2
MEDIAN is 4.5
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
20
Definitions
 Mode
the score that occurs most frequently
Bimodal
Multimodal
No Mode
the only measure of central tendency that can be
used with nominal data
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
21
Examples
a.
b.
5 5 5 3 1 5 1 4 3 5
2 2 2 3 4 5 6 6 6 7 9
c.
2 3 6 7 8 9 10
• Mode is 5
• Bimodal
• No Mode
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
22
Examples
a.
b.
5 5 5 3 1 5 1 4 3 5
2 2 2 3 4 5 6 6 6 7 9
c.
2 3 6 7 8 9 10
d.
2 2 3 3 3 4
e.
2 2 3 3 4 4 5 5
• Mode is 5
• Bimodal
• No Mode
• Mode is 3
• No Mode
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
23
Definitions
 Midrange
the value halfway between the highest
and lowest scores
Midrange =
highest score + lowest score
2
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
24
Round-off rule for
measures of central tendency
Carry one more decimal place than is present
in the orignal set of data
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
25
An Example of Skewness
Dataset 1:
Frequency
3
3, 4, 4, 5, 5, 5, 6, 6, 7
Mean = 5, Median = 5
2
Symmetric
1
0
3
4
5
6
7
C1
3
3, 4, 4, 5, 5, 5, 7, 7 ,9.
Frequency
Dataset 2:
Mean=5.444, Median = 5.
Skewed
right
2
1
0
3
4
5
6
7
8
9
C2
Dataset 3: 2, 3, 3, 5, 5, 5, 6, 6, 7.
Frequency
Mean = 4.667, Median = 5.
3
2
Skewed
left
1
0
2
3
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
4
5
C3
6
7
26
Skewness
Figure
2-8 (b)
Mode
=
Mean
=
Median
SYMMETRIC
Mean
Mode
Median
Figure
2-8 (a)
SKEWED LEFT
(negatively)
Mean
Mode
Median
SKEWED RIGHT
(positively)
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
Figure
2-8 (c)
27
Best Measure
of Central Tendency
Table 2-6
• Advantages - Disadvantages
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
28
Mean from a Frequency Table
use class mark of classes for variable x
S (f • x)
x =
Formula 2-2
Sf
x = class mark
f = frequency
Sf=n
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
29
Quiz
Scores
Frequency
Class Marks
0-4
2
2
5-9
5
7
10 - 14
8
12
15 - 19
11
17
20 - 24
7
22
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
Mean of
this
frequency
table =14.4
30
Measure of Variation
Range
lowest
score
highest
score
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
31
Measure of Variation
Standard Deviation
a measure of variation of the scores
about the mean
(average deviation from the mean)
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
32
Sample Standard Deviation
Formula
S=
S (x – x)
n–1
2
Formula 2 -4
calculators can calculate sample standard
deviation of data
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
33
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 x =104,
2
2) Jefferson Valley Bank: S x =513.27, S x =71.5, s=0.48.
2
3) Bank of Providence:
S x2 =541.09, S x =71.5, s=1.82.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
34
Population Standard Deviation
s =
S (x – µ)
N
2
calculators can calculate the
population standard deviation
of data
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
35
Symbols
for Standard Deviation
Sample
Textbook
Some graphics
calculators
Some
nongraphics
calculators
Population
s
s
Sx
xsn–1
sx
xs n
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
Book
Some
graphics
calculators
Some
nongraphics
calculators
36
Measure of Variation
Variance
standard deviation squared
}
Notation
s
s
2
2
use square key
on calculator
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
37
Variance
S (x – x) Sample
s =
n – 1 Variance
2
2
S (x – µ) Population
s=
Variance
N
2
2
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
38
Round-off Rule
for measures of variation
Carry one more decimal place than
was present in the original data
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
39
Standard Deviation
Shortcut Formula
n (S x ) – (S x)
n (n – 1)
2
s=
2
Formula 2 - 6
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
40
Same Means (x = 4)
Different Standard Deviations
FIGURE 2-10
Frequency
s=0
7
6
5
4
3
2
s = 0.8
s = 1.0
s = 3.0
1
1 2 3 4 5 6 7
1 2 3 4 5 6 7
1 2 3 4 5 6 7
1 2 3 4 5 6 7
Standard deviation gets larger as spread of data increases.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
41
FIGURE 2-10
The Empirical Rule
(applies to bell shaped distributions)
68% within
1 standard deviation
0.340
x–s
0.340
x
x+s
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
42
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
x – 2s
0.135
x–s
x
x+s
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
x + 2s
43
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
x – 3s
x – 2s
0.135
x–s
x
x+s
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
x + 2s
x + 3s
44
Range Rule of Thumb
(minimum) x –
2s
x + 2(maximum)
s
x
Range  4s
or
s
Range
4
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
45
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.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
46
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.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
47
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?
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
48
Quartiles
Q1, Q2, Q3
divides ranked scores into four equal parts
25%
25%
25% 25%
Q1 Q2 Q3
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
49
Percentiles
• 99 Percentiles
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
50
Finding the Percentile of a Given Score
Percentile of score x =
number of scores less than x
• 100
total number of scores
Sorted Axial Loads of 175 Aluminum Cans
[1] 200 201 204 204 206 206 208 208 209 215 217 218 220 223 223
[16] 225 228 230 230 234 236 241 242 242 248 250 251 251 252 252
[31] 254 256 256 256 257 257 258 259 259 260 261 262 262 262 262
[46] 262 263 263 263 263 263 264 265 265 265 266 267 267 268 268
[61] 268 268 268 268 268 268 268 269 269 269 269 270 270 270 270
[76] 270 270 270 270 271 271 272 272 272 272 272 273 273 273 273
[91] 273 273 274 274 274 274 275 275 275 275 276 276 276 276 276
[106] 277 277 277 277 277 277 277 277 278 278 278 278 278 278 278
[121] 279 279 279 280 280 280 281 281 281 281 282 282 282 282 282
[136] 282 283 283 283 283 283 283 284 284 284 284 285 285 285 286
[151] 286 286 286 287 287 288 289 289 289 289 289 290 290 290 291
[166] 291 292 292 292 293 293 294 295 295 297
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
51
Start
Finding the Value
of the
kth Percentile
Rank the data.
(Arrange the data in
order of lowest to
highest.)
Compute
L= k
n
100
(
)
where
n = number of scores
k = percentile in question
Is
L a whole
number
?
No
Yes
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.
Change L by rounding
it up to the next
larger whole number.
The value of Pk is the
Lth score, counting
from the lowest
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
52
Sorted Axial Loads of 175 Aluminum Cans
[1] 200 201 204 204 206 206 208 208 209 215 217 218 220 223 223
[16] 225 228 230 230 234 236 241 242 242 248 250 251 251 252 252
[31] 254 256 256 256 257 257 258 259 259 260 261 262 262 262 262
[46] 262 263 263 263 263 263 264 265 265 265 266 267 267 268 268
[61] 268 268 268 268 268 268 268 269 269 269 269 270 270 270 270
[76] 270 270 270 270 271 271 272 272 272 272 272 273 273 273 273
[91] 273 273 274 274 274 274 275 275 275 275 276 276 276 276 276
[106] 277 277 277 277 277 277 277 277 278 278 278 278 278 278 278
[121] 279 279 279 280 280 280 281 281 281 281 282 282 282 282 282
[136] 282 283 283 283 283 283 283 284 284 284 284 285 285 285 286
[151] 286 286 286 287 287 288 289 289 289 289 289 290 290 290 291
[166] 291 292 292 292 293 293 294 295 295 297
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.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
53
Interquartile Range:
Q3 – Q1
Semi-interquartile Range: Q3 – Q1
2
Midquartile: Q1 + Q3
2
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
54
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
55
Exploratory Data Analysis
Traditional Statistics
 Used to explore data at a
preliminary level
 Used to confirm final conclusions
about data
 Few or no assumptions are made
about the data
 Typically requires some very
important assumptions about the
data
 Tends to evolve relatively simple
calculations and graphs
 Calculations are often complex, and
graphs are often unnecessary
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
56
Boxplots
Box-and-Whisker Diagram
5 - number summary
 Minimum
 first quartile Q1
 Median
 third quartile Q3
 Maximum
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
57
Boxplots
Box-and-Whisker Diagram
60
68.5
78
90
52
Figure 2-13 Boxplot of Pulse Rates (Beats per minute) of Smokers
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
58
Figure 2-14
Normal
Boxplots
Uniform
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
Skewed
59
Outliers
Values that are very far away from most of the data
300
290
Axial Load
280
270
260
250
240
230
220
210
200
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
60
Class Survey Data
75
Height
70
65
60
n
y
Bone
Boxplots for the heights of those who never broke a
bone and those who did
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
61
When comparing two or more boxplots,
it is necessary to use the same scale.
100
PULSE
90
80
70
60
50
40
2
1
(yes)
SMOKE
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
(No)
62