Download Lab 5: Descriptive Statistics

Page 1
Technical Math II
Lab 5: Descriptive Stats
Lab 5: Descriptive Statistics
Name
Name
Name
Name
Purpose: To gain experience in the descriptive statistical analysis of a large (173 scores) data set. If you
choose you can do this lab as an Excel spreadsheet. There are some Excel hints at the end of the lab.
The placement scores of MATC students enrolled in Elementary Algebra were as follows:
36 39 29 37 50 32 38 36 34 32 32 45 32 45 34 31 48 48 54 38 44 44 38 38 39 42 44 42 44 36 36 29 38 47
44 52 31 36 40 48 32 34 49 31 34 38 31 38 36 32 42 40 32 34 44 38 47 44 36 40 38 34 36 45 38 49 34 23
40 47 32 47 38 38 34 36 32 48 38 48 32 23 42 34 47 42 36 38 29 42 31 29 42 31 29 36 34 36 44 45 36 38
44 42 29 34 44 42 34 34 36 34 38 40 38 29 42 38 51 48 34 44 48 38 38 34 42 47 44 34 27 34 36 38 40 32
29 44 34 42 44 49 47 32 40 42 26 36 34 45 38 44 36 45 36 40 42 42 48 27 36 32 45 44 36 42 36 27 48 40
38 42 38
First organize the data into a stem-and-leaf diagram. For each stem use two lines of leaves. On the top
line place the leaf digits 0 through 4 and on the bottom line the leaf digits 5 through 9.
Stem
Leaves
2
3
4
5
Next organize the data into a frequency distribution and enter the results into either Table 1 or an Excel
spreadsheet. This full data set will be referred to as the 'Ungrouped' Data. For this set of scores calculate
and record to the nearest thousandth the descriptive statistics requested in the left side of Table 3. These
calculations can all be done using Excel formulas. See the Excel hints at the end of the lab. Now, group
the data so that the lowest class is 20.5-24.5. Fill in Table 2. Using this grouped data, construct a
histogram of the scores, plotting either frequency or relative frequency along the vertical axis and the
classes along the horizontal axis. Use the midpoint or class mark of each class to represent all of the
scores in that class and repeat the same calculations as for the Ungrouped Data. Fill in the right side of
Table 3, labeled as Grouped Data .Finally, make a box plot of both the Ungrouped and Grouped data.
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Technical Math II
Lab 5: Descriptive Stats
Table 1
Score
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
f
Relative f
Cumulative f
Rel. Cum. f
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Technical Math II
Lab 5: Descriptive Stats
Table 2
Class
20.5 – 24.5
24.5 – 28.5
28.5 – 32.5
32.5 – 36.5
36.5 – 40.5
40.5 – 44.5
44.5 – 48.5
48.5 – 52.5
52.5 – 56.5
f
Class Mid Point
Relative f
Cumulative f
22.5
Histogram of Grouped Data
Rel. Cum. f
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Technical Math II
Lab 5: Descriptive Stats
Table 3
Descriptive Statistic
Ungrouped Data
Minimum (smallest score)
Maximum (largest score)
Range = Max – Min
Mode
Median Md
Mean x
Lower Quartile Q1
Upper Quartile Q3
Interquartile Range
Sample Standard Deviation sx
Population Standard Deviation σ x
Box Plot of the Ungrouped Data
Box Plot of the Grouped Data
Grouped Data
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Technical Math II
Lab 5: Descriptive Stats
Discussion Questions:
How closely do the descriptive statistics of the grouped and ungrouped scores compare?
How well does grouping the scores into classes represent the actual data?
For the ungrouped data, what fraction of the scores is within one standard deviation of the mean?
For the ungrouped data, what fraction of the scores is within two standard deviations of the mean?
For the ungrouped data, what fraction of the scores is within three standard deviations of the mean?
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Technical Math II
Lab 5: Descriptive Stats
Hints for Using Excel
Excel’s has built in statistical functions that can perform many of the calculations for this lab. To access
them right click on the fx in the formula bar, then select the Statistical category. Each function gives
details as to what it calculates and how to enter the data cells.
To construct a histogram choose the following Column Chart sub-type.
To close the gap between the rectangles of the histogram right click on one of the rectangles and choose
the Format Data Series. Set the Gap width to zero.
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Technical Math II
Lab 5: Descriptive Stats
When working with a frequency distribution the mean and standard deviations should be calculated using
the column of scores and the column of frequencies and the following formulas:
x=
∑f x
i
n
i
=
∑f x
∑f
i
i
;
i
(∑ f x )
−
2
s=
∑ f (x
i
i
− x)
n −1
2
=
∑f x
i
2
i
i
n
n −1
(∑ f x )
−
2
i
;σ =
∑ f (x
i
i
n
− μ)
2
=
∑f x
i
2
i
i
i
n
n
The needed sums can be calculated by generating x*f and x^2*f columns. This is illustrated below.
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Technical Math II
Lab 5: Descriptive Stats