Download Measures of Central Tendancy

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
Review of Descriptive Graphs and Measures
Here are some ways to display
Categorical Data:
Pie Chart
hours of detention
8.33%
Pareto Chart
2.78%
11.11%
36.11%
tardies
no id
talk
walk
throw
16.67%
lie
25.00%
Hours of Detention
frequency
14
12
Category
10
tardies
8
6
frequency
percent
cumulative
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
36
1.00%
4
Bar Graph
2
0
tardies
no id
talk
walk
offense
throw
lie
total
Review of Descriptive Graphs and Measures
Stem-and-leaf
6 |7
7 |18
8 |25677
9 |25799
10 | 0 1 2 3 3 4 5 5 7 8 9
11 | 2 6 8
12 | 2 4 5
Low/High Stem-and-leaf
6|7
7|1
7|8
8|2
8|5677
9|2
9|5799
10 | 0 1 2 3 3 4
10 | 5 5 7 8 9
11 | 2
11 | 6 8
Review of Descriptive Graphs and Measures
Dot-plot or Line Plot
Phone
66
76
86
96
minutes
106
116
126
Review of Descriptive Graphs and Measures
Min
Q1
median Q3
42
30
45
17
15
max
55
25
35
45
55
Interquartile Range = 45 – 30 = 15
Review of Descriptive Graphs and Measures
Absences
x
8
2
5
12
15
9
6
Scatterplot
Final
grade
(y)
95
90
85
80
75
70
65
60
55
50
45
40
0
2
4
6
8
10
12
Absences (x)
14
16
Grade
y
78
92
90
58
43
74
81
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.
Measures of Central Tendency
We can also find the mean of a frequency distribution
xf . This is usually easier
by calculating
mean 
to do with a table:
n
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
Measures of Central Tendency
For classes containing multiple values, you use the
midpoint of the class as the x.
xf
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 
n
Mean =66/14  4.71
Measures of Central Tendency
The class with the highest frequency is called the
modal class.
Class
midpoint
f
xf
modal class
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
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.
Measures of Central Tendency
Finally, there is the weighted mean:
x
weight
xw
86
.5
43
Classwork 90
/homewk
.25
22.5
Quizzes
.25
19
Tests
76
84.5
xw
mean 
n
Measures of Variation
• The range is the largest value minus
the smallest value
• The Interquartile range is the Third
Quartile minus the First Quartile
Measures of Variation
The Variance is :
2

(
x


)
2 
n
Example data set one: 1, 3, 5, 7, 8, 9, 9, 11, 12, 12, 15
The mean is about 8.36
The variance is [(1-8.36)2+(3-8.36)2+(5-8.36)2+(78.36)2+(8-8.36)2+(9-8.36)2+(9-8.36)2+(11-8.36)2+(128.36)2+(12-8.36)2+(15-8.36)2]/11=3.98
The Standard Deviation is the square root of the
Variance.
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 L1
Press Stat. Cursor right to choose Calc. Enter for one
variable stats.
The mean, standard deviation and several other
measures will be displayed.
Measures of Variation
The standard deviation can be calculated using
a table.
x f
xf
2 
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
2
n
(
n
)2
222 54 2
 
( )
14
14
 2  .9796
  .9897
2
Measures of Variation
However the calculator is still probably easier:
x
f
2
1
3
4
4
6
5
2
6
1
14
TI-83
Enter values in L1
Enter frequencies in L2
One-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.
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
120
100
percentile
80
60
40
20
0
0
20
40
60
age
80
100
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
Measures of Variation
Z-score
The number of standard deviations a data point
is from the mean.