math tech iiii, nov 11...mean is used to compute other statistics, such as variance and standard...
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Math Tech IIII, Nov 11
Measures of Variation III – Chebychev’s
Theorem and Understanding Measures of
Variation
Book Sections: 2.4 Essential Questions: How do I compute and use statistical values?
What do I do in variation when the data is skewed? How do I apply
measures of variation?
Standards: DA-4.5, DA-4.6, DA-4.7, DA-4.9, DA-4.10, S.ID.1, .2, .3, .4
Empirical Rule Words and Graph Together
• 68% of the data lies within
one standard deviation of the
mean
• 95% of the data lies between
two standard deviations of the
mean
• 99.7% of the data lies
between three standard
deviations of the mean
Example 1
• A symmetric data set has a mean of 50 and a
standard deviation of 10. What percent of the
data is between 40 and 60?
Example 2
• A symmetric data set has a mean of 75 and a
standard deviation of 15. What is the range of
the middle 95% of the data?
Chebychev’s Theorem Words and Graph Together
• The portion of ANY data
set lying within k standard
deviations (k > 1) of the
mean is at least:
• k = 2: In any data set, at
least 1 – 1/22 = ¾ or 75% of
data are within 2 standard
deviations.
• k = 3: In any data set, at
lease 1 – 1/32 = 8/9 or 88.9%
of the data lie within 3
standard deviations
2
11
k
Example
• The age distributions for Alaska and Florida are shown
above. Decide which is which. Apply Chebychev’s theorem
to both and draw a conclusion.
x = 31.6
s = 19.5
x = 39.2
s = 24.8
Example
x = 31.6
s = 19.5
x = 39.2
s = 24.8
One More Look at the Skew
Skewness is caused by a significant difference between mean
and median.
Mean vs Median
• The median divides a data set in half resulting in two
equal parts. It can be a data value, but is not always.
It is not greatly affected by outliers. The median is
used to find data quartiles and outliers.
• The mean is a unique value computed from all the
data values. It is not used in set division and is not
usually a data value. It is affected by outliers. The
mean is used to compute other statistics, such as
variance and standard deviation.
Variance and Standard Deviation
• The standard deviation is the square root of the
variance.
• The Variance and standard deviation are determined
by the spread of the data. If they are large, the data is
dispersed, if they are small, the data is compact.
• Variance and standard deviation are used to
determine the consistency of a variable. In
manufacturing, the variance of parts must be within a
certain tolerance or parts will not fit together.
Variability – The Concept
• Variability is how spread out a set of data is.
• In comparing two data sets to see which one is more
variable – compute standard deviations. The one with
the largest s is more spread out and is said to be more
variable.
• You can sometimes see variability in a data set. You
can always compute and compare s’s.
Example
• Two brands of paint are tested for durability in
fading with the following results in months. Each has
a mean of 35 months. Which brand is more
consistent?
Brand A Brand B
10 35
60 45
50 30
30 35
40 40
20 25
Examples
The average daily high temps for January for 10 selected cities is:
50, 37, 29, 54, 30, 61, 47, 38, 34, 61
And their normal monthly precipitation for January is:
4.8, 2.6, 1.5, 1.8, 1.8, 3.3, 5.1, 1.1, 1.8, 2.5
Which set is more variable?
Variability by Sight
Which data set is more variable:
0 4
1 5 7
2 3 3 5 9
3 1 3 7 9
4 0 3 6 8 9
5 1 5 7 8
6 0 3 6 8
7 0 7
8 2
Key 1|2 = 12 1 8
2 3
3 1 7
4 1 2 5
5 1 1 3 4 5 5 5 6 7 7 7 8 9
6 4 4 8
7 3 7
8 0
9 5
Variability by Sight
Which data set is more variable:
Class work: Classwork Handout 1-10
Homework: None
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