review of descriptive graphs and measures here is a quick review of what we have covered so far. pie...
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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
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Review of Descriptive Graphs and Measures
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%
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Review of Descriptive Graphs and Measures
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
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Review of Descriptive Graphs and Measures
66 76 86 96 106 116 126
Phone
minutes
Dot-plot or Line Plot
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Review of Descriptive Graphs and Measures
5545352515
42 453017 55
Interquartile Range = 45 – 30 = 15
Min Q1 median Q3 max
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Review of Descriptive Graphs and Measures
Absences Grade
0 2 4 6 8 10 12 14 16
404550556065707580859095
Absences (x)
x825
121596
y78929058437481
Finalgrade
(y)
Scatterplot
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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.
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Measures of Central Tendency
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
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Measures of Central Tendency
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
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Measures of Central Tendency
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
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Measures of Central Tendency
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.
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Measures of Central Tendency
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
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Measures of Variation
• The range is the largest value minus the smallest value
• The Interquartile range is the Third Quartile minus the First Quartile
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Measures of Variation
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
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Measures of Variation
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.
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Measures of Variation
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
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Measures of Variation
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
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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.
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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.
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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)
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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
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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
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Measures of Variation
Z-score
The number of standard deviations a data point is from the mean.