Review of Descriptive Graphs and Measures Here is a quick review of what we have covered so far. Pie...

Review of Descriptive Graphs and Measures

Here is a quick review of what we have covered so far.

•Pie Charts•Bar Charts•Pareto•Tables•Dotplots•Stem-and-leaf•Histograms•Ogives•Boxplots•Time Series

•Mean •Median •Mode •Weighted mean •Range •IQR •Variance and Standard Deviation •Mean and Standard Deviation of a frequency distribution •Median of a distribution•Empirical Rule •Z-scores •Quartiles, percentiles


Here are some ways to display Categorical Data:

Pareto Charthours of detention

36.11%

25.00%

16.67%

11.11%

8.33%2.78%

tardies

no id

talk

walk

throw

lie

Hours of Detention

0

2

4

6

8

10

12

14

tardies no id talk walk throw lie

offense

freq

uen

cy

Bar Graph

Pie Chart

Category frequency percent cumulative

tardies 13 36.11% 13

no id 9 25.00% 22

talk 6 16.67% 28

walk 4 11.11% 32

throw 3 8.33% 35

lie 1 2.78% 36

total 36 1.00%


6 | 7

7 | 1 8

8 | 2 5 6 7 7

9 | 2 5 7 9 9

10 | 0 1 2 3 3 4 5 5 7 8 9

11 | 2 6 8

12 | 2 4 5

Stem-and-leaf Low/High Stem-and-leaf

6 | 7 7 | 1

7 | 8 8 | 2 8 | 5 6 7 7 9 | 2 9 | 5 7 9 9 10 | 0 1 2 3 3

4 10 | 5 5 7 8 9 11 | 2 11 | 6 8


66 76 86 96 106 116 126

Phone

minutes

Dot-plot or Line Plot


5545352515

42 453017 55

Interquartile Range = 45 – 30 = 15

Min Q1 median Q3 max


Absences Grade

0 2 4 6 8 10 12 14 16

404550556065707580859095

Absences (x)

x825

121596

y78929058437481

Finalgrade

(y)

Scatterplot

Measures of Central Tendency

•The mode is the value that occurs the most. There can be more than one mode.

•The median is the middle value in an ordered data set

•The arithmetic mean is the center of gravity of the data set. This is obtained by summing all of the values and dividing by the number of values.


We can also find the mean of a frequency distribution

by calculating . This is usually easier to do with a table: n

xfmean

x f xf

2 1 2

3 4 12

4 6 24

5 2 10

6 1 6

14 54

Mean = 54/14 3.86


For classes containing multiple values, you use the midpoint of the class as the x.

n

xfmean

Class midpoint f xf

0-1.9 1 1 1

2-3.9 3 4 12

4-5.9 5 6 30

6-7.9 7 2 14

8-9.9 9 1 9

14 66

Mean =66/14 4.71


The class with the highest frequency is called the modal class.

modal classClass midpoint f xf

0-1.9 1 1 1

2-3.9 3 4 12

4-5.9 5 6 30

6-7.9 7 2 14

8-9.9 9 1 9

14 66


We estimate the median as the midpoint of the class it lies in.

Class midpoint f xf

0-1.9 1 1 1

2-3.9 3 4 12

4-5.9 5 6 30

6-7.9 7 2 14

8-9.9 9 1 9

14 66

median lies in here, so we estimate the median as 5.


Finally, there is the weighted mean:

n

xwmean

x weight xw

Tests 86 .5 43

Classwork/homewk

90 .25 22.5

Quizzes 76 .25 19

84.5

Measures of Variation

• The range is the largest value minus the smallest value

• The Interquartile range is the Third Quartile minus the First Quartile


The Variance is :

Example data set one: 1, 3, 5, 7, 8, 9, 9, 11, 12, 12, 15The mean is about 8.36The variance is [(1-8.36)2+(3-8.36)2+(5-8.36)2+(7-8.36)2+(8-8.36)2+(9-8.36)2+(9-8.36)2+(11-8.36)2+(12-8.36)2+(12-8.36)2+(15-8.36)2]/11=3.98The Standard Deviation is the square root of the Variance.

22 ( )x

n


The standard deviation is the easier to find using a calculator with the function built in.

Example data set one: 1, 3, 5, 7, 8, 9, 9, 11, 12, 12, 15

TI-83: Put the data in L1Press Stat. Cursor right to choose Calc. Enter for one variable stats.The mean, standard deviation and several other measures will be displayed.


The standard deviation can be calculated using a table.

x f xf x2f

2 1 2 4

3 4 12 36

4 6 24 96

5 2 10 50

6 1 6 36

14 54 222

22 2( )

x f xf

n n

2 2

2

222 54( )

14 14

.9796

.9897


However the calculator is still probably easier:

x f

2 1

3 4

4 6

5 2

6 1

14

TI-83Enter values in L1Enter frequencies in L2One-variable-stats L1, L2

This will give you the standard deviation of the frequency table

Empirical Rule

The Empirical Rule for Normal Distributions

About 68% of all values fall within 1 standard deviation of the mean About 95% of all values fall within 2 standard deviation of the mean

About 99.7% of all values fall within 3 standard deviation of the mean.

Empirical Rule

The Empirical Rule for Normal Distributions

About 68% of all values fall within 1 standard deviation of the mean About 95% of all values fall within 2 standard deviation of the mean

About 99.7% of all values fall within 3 standard deviation of the mean.

Example: A normal dataset has a mean of 50 and a standard deviation of 5. Between what two numbers does 95% of the data fall?

(50-2*5, 50+2*5) (40, 60)

Percentiles

•Count the number of data points that lie below the value•Divide this by the total number of data points•Convert to a percent (multiply by 100)

Reading a percentile chart:

Age of Executives

0

20

40

60

80

100

120

0 20 40 60 80 100

age

perc

en

tile

Z-scores

Z-score:

The number of standard deviations a data point is from the mean.

Find the raw distance from the mean.Divide by the standard deviation.

Example: A data set has a mean of 50 and a SD of 5.What is the z-score of 62?

Z = (x – mean)/SD = (62 – 50)/5 = 12/5 = 2.4


Z-score

The number of standard deviations a data point is from the mean.

Review of Descriptive Graphs and Measures Here is a quick review of what we have covered so far. Pie...

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