1
Chapter 3: Descriptive Measures
Statistics 211: Tu-Th, 9/13-15/2011
3.1 Measures of Center
Mean of a Data Set (with single variable) –
sum of the observations divided by the number of
observations
Median of a Data Set (with single variable) –
the middle of the data set; if odd then middle of
the ordered list; if even then the average or mean or the two middle points
Mode of a Data Set – a value that occurs more than once or greatest frequency; if does not occur
more than once then no mode
Comparison of Mean, Median and Mode
Mean
=
Sum of Observations
No. of Observations
Median =
Middle Value in Ordered List
Mode
=
Most frequent value
Where (relative positions) do mean and median fall in right skewed, symmetric or left
skewed?
Resistant Measure –
sensitive to influence of a few extreme observations;
median
is a resistant
measure of center; but the
mean
is not. Why?
Trimmed Mean –
removing portion of extreme values before calculating average
Examples when to use Mean (test scores), Median (Real Estate Price), Mode (Boston
Marathon Runners)
Sample Mean
±±
x
g
=
±
∑ g
g
G
Test Score Example:
Sum or
∑
X
i
= X
1
+ X
2
+ X
3
+ X
4
= 88 + 75 + 95 + 100
=
358/4 = 89.5 average test score
3.2 Measures of Variation
Range of Data Set
– distance between smallest and largest endpoint values of data set
Range = Maximum value – Minimum Value
Sample Standard Deviation
– how far on average the observations are from the
mean

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*2
Deviations from the Mean - Example
Height Deviation from the Mean
Standard Deviation
x
x –
gG
______________________( x –
)
2
______
72
-3
0
73
-2
4
76
1
1
76
1
1
78
3
9
∑
24
Sum of Squared Deviations
– subtract mean from each observation; then square each value
Sample Variance
–
±
g
²
=
²
∑
±
(
G
g±
–
±G̅
)
G
²±³±´
= 24/5-1 = 6
(dividing by n-1 because estimating sample variance from population variance)
Sample Standard Deviation – Defining Formula
–
how far on average the observations in the
sample are from the mean of the sample (square root of standard deviation)
s =
³∑
(
G
g
³G̅
)
g
²³´
=
√
6
= 2.4 inches
height of starting players
Example Starting Height of Players (Example No. 2): 67, 72, 76, 76, 84
Step 1: Compute Mean
²²
x
´
=
²
∑ µ
g
²
= 75 inches
Step 2: Construct table to obtain sum or squared deviations = 156
Step 3: Calculate sample standard deviation
s =
³∑
(
G
g
³G̅
)
g
²³´
=
µ
156/4
= 6.2 Heights of the players vary from the mean of 75 inches by 6.2

This is the end of the preview.
Sign up
to
access the rest of the document.