Download Lecture 3 - UC Davis Statistics

STA 13
Lecture 3
Descriptive Statistics (continued)
2. Measures of Dispersion
The sample mean and median does not give us information about the
spread of the data.
Example: Consider grades of two classes:
Class 1
48
48
50
Class 2
51
53
11
39
50
55
The average deviation from the sample mean is always 0.
a. Range : Difference between the largest and smallest observations.
Example 1:
Class 1:
Class 2:
Example 2: Ozone data.
Range = 9.4 – 1.4 = 8.0
95
b. Sample Variance: The average (almost) of squared deviations from the
mean.
_
2
s =  ( x – x )2 / (n-1)
c. Sample Standard Deviation: The square root of variance
_______________
s =   ( x – x-)2 / (n-1)
Example 1:
Class 1
Class 2
x
x
48
48
50
51
53
11
39
50
55
95
Example 2: Ozone data
s2 = 3.6842
s =  3.6842 = 1.920
d. Coefficient of Variation: Scale free measure of variation
_
Sample C.V. = s / x
Example: Ozone data
C.V. = 1.92/4.612 = 0.4162 or approximately 42%
In applications if CV is more than 60% the variation is large.
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e. Empirical Rule:
For large data, 68% of data fall within one standard deviation of the
mean., and 95% within two standard deviations.
_______________________|______________________
mean
f. Interquartile Range: The difference between the third and fist quartiles.
IQR = Q3 – Q1
Example: Ozone data
IQR = 5.7 – 3.5 = 2.2
Boxplot
-----------------------------I
+
I------------*
-----------------+---------+---------+---------+---------+---------+---4.5
6.0
7.5
9.0
g. The z-Score: Measure of relative standing.
z-score = (x-mean) / standard deviation
Example: Grades in two classes
Class 1: mean = 50
Score = 51
Class 2: mean = 50
Score = 65
standard deviation = 2
z-score =
standard deviation = 30
z-score =
3
Planning and Conducting a Study
In planning and conducting a study we need to:
 Understand the problem
 Decide what to measure
 Collect the data
 Use descriptive methods to summarize the data
 Use statistical methods to analyze
 Interpret the results
 Discuss
Examples:
 A medical study
 A study in environmental pollution
 People’s opinion on social problems
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