251solnG3 1/31/08
G. Measures of Dispersion and Asymmetry.
1. Range
Downing & Clark, problem 7 above (Use data to find IQR). Review solutions and terms on page 41 (36 in 3rd ed.) of Downing & Clark.
2. The Variance and Standard Deviation of Ungrouped Data.
Text exercises 3.1b, 3.2b, 3.6, 3.37, 3.24 [3.1b, 3.2b, 3.7, 3.37, 3.23] (3.1b, 3.2b, 3.7, 3.23, 3.33)
3. The Variance and Standard Deviation of Grouped Data.
Text exercises 3.28, 3.30 (3.68, 3.70) (work 3.30 in thousands), Downing & Clark pg 42 or 37, problems 6,7 (Find sample standard deviation – hint: run problem 6 in hundreds) (Note that you can use the Excel or Minitab techniques in the graded assignment to compute and sum the and columns in problems 6 and 7. ), Problems G1, G2. Graded Assignment 1
4. Skewness and Kurtosis.
Find the standard deviation, coefficient of variation and measures of skewness in Text problem 3.1, 3.2. Problems G3A, G4 (See 251wrksht).
5. Review
a. Grouped Data.
b. Ungrouped Data.
Part of Section 4 is in this document.
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Problem G3A: Use computational formulas on the data below. Consider the data a sample.
a. Complete the cumulative frequency under .
b. Calculate the mean.
c. Calculate the median.
d. Calculate the mode.
e. Calculate the variance.
f. Calculate the interquartile range.
g. Calculate the standard deviation.
h. Calculate a statistic showing skewness.
i. Show all the data presented on a histogram with six class intervals.
j. Put a box plot below the histogram.
Now repeat Problem G3A using definitional formulas.
0 - 9.999 / 5010 - 19.999 / 50
20 - 29.999 / 100
30 - 39.999 / 150
40 - 49.999 / 50
Solution: Fill in the table. Note that the conventional way of writing the headings is Class, , , F, , and . We use computational formulas first. So if x = 5 and f = 50, , and .
/(midpoint) / / / / /
0 - 9.999 / 5 / 50 / 50 / 250 / 1250 / 6250
10 - 19.999 / 15 / 50 / 100 / 750 / 11250 / 168750
20 - 29.999 / 25 / 100 / 200 / 2500 / 62500 / 1562500
30 - 39.999 / 35 / 150 / 350 / 5250 / 183750 / 6431250
40 - 49.999 / 45 / 50 / 400 / 2250 / 101250 / 4556250
400 / 11000 / 360000 / 12725000
To summarize our results, , and .
a. Complete the cumulative frequency under F: (See above.) We add down the column.
b. Calculate the mean:
c. Calculate the median: To get a measure of position in grouped data
first use , then use to find the value. Here
. So . This location is above 200 and below 350, so use 20
to 29.9999. Then .
d. Calculate the mode:
The group 30-39.999 has a frequency of 150, which is the largest frequency. So the mode is
35.00, its midpoint.
e. Calculate the variance:
f. Calculate the interquartile range:
For the first quartile . This location is above 100 and
below 200, so use 20 to 29.999. Then, using we find
For the third quartile . This location is above 200 and below 350, so
use 30 to 39.999. Then, we find . So
g. Calculate the standard deviation: . Note also
the coefficient of variation .
h. Calculate a statistic showing skewness: There are three possibilities:
1)
.
2) .
3) Pearson’s Measure of Skewness .
Only one of these three is needed. All indicate skewness to the left.
i. Show all the data presented on a histogram with five class intervals.
j. Put a box plot below the histogram. The box will begin at 20 and end at 36.67 with a band at
30 to indicate the median.
(Include a hand-drawn solution to i and j.)
Now we do the problem using definitional formulas. Note how much bigger the table has to be! Once again, the conventional headings would be Class, x, f, fx, , , and . There is no reason to use both the computational and the definitional methods unless specifically requested, though, of course, one method serves as a check on the other.
/(midpoint) / / / / / / /
0 - 9.999 / 5 / 50 / 250 / 50 / -22.5 / -1125 / 25312.5 / -569531.25
10 - 19.999 / 15 / 50 / 750 / 100 / -12.5 / -625 / 7812.5 / -97656.25
20 - 29.999 / 25 / 100 / 2500 / 200 / -2.5 / -250 / 625.0 / -1562.50
30 - 39.999 / 35 / 150 / 5250 / 350 / 7.5 / 1125 / 8437.5 / 63281.25
40 - 49.999 / 45 / 50 / 2250 / 400 / 17.5 / 875 / 15312.5 / 267968.75
400 / 11000 / 0 / 57500.0 / -337500.00
We can summarize the table as follows: , .
e. Calculate the variance:
h. Calculate a statistic showing skewness:
Other calculations are the same as on the previous page.
Problem G4: For the sample below, compute the following:
b. the mean
c. the median (hint: put in order first!)
d. the mode
e. the variance
f. the interquartile range
g. the standard deviation
h. a statistic showing skewness.
Solution: Both the computational and definitional method are shown. There is no reason to do both unless specifically requested.
Computational Method / Definitional Method / x in order1 / 1 / 1 / -2.75 / 7.5625 / -20.79688 / 1 / 1
2 / 4 / 8 / -1.75 / 3.0625 / -5.35938 / 2 / 1
4 / 16 / 64 / 0.25 / 0.0625 / 0.01563 / 3 / 2
5 / 25 / 125 / 1.25 / 1.5625 / 1.95313 / 4 / 2
6 / 36 / 216 / 2.25 / 5.0625 / 11.39063 / 5 / 3
3 / 9 / 27 / -0.75 / 0.5625 / -0.42188 / 6 / 3
3 / 9 / 27 / -0.75 / 0.5625 / -0.42188 / 7 / 3
7 / 49 / 343 / 3.25 / 10.5625 / 34.32813 / 8 / 4
8 / 64 / 512 / 4.25 / 18.0625 / 76.76563 / 9 / 5
3 / 9 / 27 / -0.25 / 0.5625 / -0.42188 / 10 / 6
1 / 1 / 1 / -2.75 / 7.5625 / -20.79688 / 11 / 7
2 / 4 / 8 / -1.75 / 3.0625 / -5.35438 / 12 / 8
45 / 227 / 1359 / 0.00 / 58.2500 / 70.87499
So , ,, and
b. the mean:
c. the median (hint: put in order first!):
To get a measure of position first use .
This implies that we want the mean of and . . We can also use the method for finding any fractile.. From this we get and
. Now use . So . The two ways of finding the median of ungrouped data always give identical results.
d. the mode: The mode is 3, since that appears most.
e. the variance:
i) Computational Formula
ii) Definitional Formula
You need only one of these two. I strongly recommend the first one.
f. the interquartile range:
First Quartile . From this we get and
. Now use . So
.
Third Quartile . From this we get and
. Now use . So
.
g. the standard deviation: .
Note also the coefficient of variation .
h. a statistic showing skewness. There are three possibilities:
i) Computational Formula for Skewness
.
ii) Definitional Formula for Skewness
.
iii) Relative Skewness .
iv) Pearson’s Measure of Skewness .
Only one of these four is needed. All of these are positive, indicating skewness to the right.
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