Download Notes Organizing and Describing Data

Notes Organizing and Describing Data
Univariate Data
Bivariate/Multivariate Data
Qualitative Data (Categorical)
Quantitative Data (Numerical)
2 types of Quantitative Data
1. Discrete –
2. Continuous –
Frequency vs. Relative Frequency
Types of Displays
See other handout: Bar Graphs, Pie Charts, Dotplots, Stemplots, Histograms,
Time Plots, Boxplots, Scatterplots, Ogives
Describing the overall pattern of a distribution
1. Center
2. Unusual Features
3. Shape
4. Spread
Dotplots
 A dotplot is created by using a portion of a horizontal real number line (WELL LABELLED) – no vertical axis
 Each data value is represented in the graph by a single dot above the line at its value
 If the same value appears more than once, the dots should be stacked such that stacks with the same number
of “dots” are the same height
 Dotplots work best for small discrete data sets with a moderately small spread.
Example: Test Scores
95, 96, 90, 95, 88, 95, 97, 89, 92, 95, 94, 94, 96, 95, 94, 93, 94
Stemplots
 Also called a stem-and-leaf plot
 Formed by separating each data value into two parts: one called the stem and the other, the leaf. Stems may
consist of more than one digit while leaves always consist of a single digit. The leaf is always the last place
value digit used from the original data (data is sometimes rounded to minimize the number of stems).
 To construct a stemplot, the stems (and any “missing” values in the interval of the stems) are arranged
vertically, with the smallest stem at the top and the largest at the bottom. Leaves are placed to the right of
the corresponding stem: they should be arranged in order from smallest to largest with no commas between
leaves.
 Generally, you want to have between 5 and 10 stems (including stems with no leaves). If you have too many, you
can round your data to shorten the number of stems; too few, you can “split” your stems (will see in example…).
Stems should always be split so that they each hold an equal range of values [i.e. if one stem holds GPAs of 3.0
to 3.3 (4 possible values), you can’t have another stem holding GPAs of 3.4 to 3.6 (3 possible values)].
 You must also be sure to include a “legend” with your stemplot which indicates what your original values looked
like (ex. where 6 | 2 represents 62 inches). Like all other graphical displays, be sure to give your overall graph
a descriptive title.
Example: Scores on a Psychological Test
154, 109, 137, 115, 152, 140, 154, 178, 200,
103, 126, 126, 137, 165, 165, 129, 200, 148
Histograms

A histogram strongly resembles a bar graph, with important differences.

Important terminology involving histograms:

Class: An interval containing data observations. Each observation from the data set must fall in one and only one class.

Class boundaries: Endpoints or limits for each class – defined to one additional decimal place than the largest number of
decimal places in the data set.

Class width: Distance between the class boundaries of a class.

Frequency of a class: The number of values from a data set that fall within a specific class. The sum of the
frequencies of all the classes should equal the number of values in the original data set.

Relative Frequency of a class: Equals the class frequency for that class divided by the number of values in the data set.
Shows the proportion of the whole data set contained within the class.

Cumulative Frequency: The sum of the frequencies for the current class and all preceding classes.

Cumulative Relative Frequency: The sum of the relative frequencies for the current class and all preceding classes.

To create a histogram:
1. Identify the smallest value in the data set (Xmin) and largest value in the data set (Xmax). You may wish to round data
values what aren’t whole numbers.
2. Determine the number of classes you will use for your histogram. The rule of thumb we will use to find the desired
number of classes is as follows:
The number of classes (k) to be used in constructing a histogram for sample data is the smallest integer value of k such
that 2k n, where n is the size of the data set. For example,
n
k
8 or less
3
9 – 16
4
17 – 32
5
33 – 64
6
3. Decide on class endpoints so that each class has the same width and every observation can be classified uniquely in
exactly one class. An appropriate class width can be found using the formula:
X max  X min
Class width =
k
This value is bumped-up (not rounded!) to the next integer value. This value is how wide each class (bar) is.
4. Create a frequency table:
First column – class limits
Second column – class boundaries, which are expanded class limits, so the bars touch.
Third column –frequency
Fourth column – relative frequencies
Fifth column – cumulative frequency
Sixth column – cumulative relative frequency
5. To actually create the histogram, do the following –
(a) On the x-axis, use class boundaries. Start at the left edge of the graph, even if the left side of your class is
negative.
(b) On the y-axis, mark either frequencies or relative frequencies, depending on what the problems asks you to do.
(c) Label both axes and title your graph!This is one of the most important aspects of graphing data.
(d) Draw your classes (bars), based on the frequencies or relative frequencies obtained in the frequency table.Since
your data is univariate (one category), the classes should touch. On a categorical graph, bars are separate because
the categories aren’t the same.
Creating a Frequency Table and a Histogram
One way Commuting Distances in Miles for 60 workers in Downtown Dallas
13
7
12
6
34
14
47
25
45
2
13
26
10
8
1
14
41
10
3
21
8
13
28
24
16
19
4
7
36
37
20
15
16
15
17
31
17
3
11
46
24
8
40
17
18
12
27
16
4
14
23
9
29
12
2
6
12
18
9
16
Number of Classes: __________________
Width of class limits:
Max - Min
Number of classes
(then bump up!) ________________
Create a Frequency Distribution for the above:
Class Limits
Class Boundaries
Frequency
Draw the histogram and then CUSS and BS it!
Relative
Frequency
Cumulative
Frequency
Cumulative
Relative Freq.
Ogives
The last two columns on the frequency table deal with what is happening on a cumulative basis.
Either one of the last two columns can be used to make an ogive (although cumulative relative frequency
proves to be more useful). Class boundaries are placed on the horizontal axis in the same manner as
with a histogram, while either cumulative or (most likely) cumulative relative frequencies are placed on
the vertical axis. Points are graphed above the upper class boundariesand are then connected with line
segments. Points/lines are used to show how much of the total data set has been “accumulated” at the
end of each class.
Note: When cumulative relative frequencies are used to create an ogive, the ogive can quickly provide
accurate estimates of a percentile values, which is the data value at which that percent of values
occurs before the stated value. Quartiles are located every 25% of the data. The first quartile
(Q1) is the 25thpercentile, the second quartile (Q2 or Median) is the 50th percentile while the
third quartile (Q3) is the 75th percentile. Interquartile range (IQR) is found by subtracting Q1
from Q3. IQR = Q3 – Q1.
Ex. Draw an ogive of One way Commuting Distances in Miles for 60 workers in Downtown Dallas. Use the
ogive to estimate the middle of the data set.
The following cumulative relative frequency plot shows the
time (in minutes) that it took students to finish quiz 1.
2) How much time did it take the fastest 15% to finish their
quiz?
3) How long did the slowest person take?
4) What percent of the students were finished after 15
minutes?
5) How many people were finished at the 22 minute mark?
Cumulative relative frequency
1) What is the median time it took to complete quiz 1?
1
.9
.8
.7
.6
.5
.4
.3
.2
.1
5
10
15
20
25
Minutes
30
35
40
Measures of Center
Mean
Median
Mode
Resistance
1. Traumatic knee dislocation often requires surgery to repair ruptured ligaments. One measure of recovery is
range of motion (measured by the angle formed when, starting with the leg straight, the knee is bent as far as
possible). The article “Reconstruction of the Anterior and Posterior Cruciate Ligaments after Knee
Dislocation” reported the following post surgical range of motion for a sample of 13 patients.
154
135
142
108
137
120
133
127
122
134
126
122
135
Find the mean, median and mode.
2. The paper “The Pedaling Technique of Elite Endurance Cyclists” reported the accompanying data on singleleg power at a high workload.
244
205
191
211
160
183
187
211
180
180
176
194
174
200

Find the mean, median and mode.

Suppose the first observation had been 204, not 244. How would the mean and median change? Which
measure would you say is nonresistant to outliers?

Calculate a trimmed mean by eliminating the smallest and largest sample observations.
3. The results of an AP Biology Leaf Disk Lab are recorded in the table below
Back to back, split-stem stemplot
Making a boxplot
Summarize
Describe each distribution and compare
Boxplots
5 number summary
IQR
Outliers
Boxplot vs. Modified Boxplot
Consumer Reports did a study of ice cream bars in their August 1989 issue. Twenty-seven bars having a
taste-test rating of at least “fair” were listed, and calories per bar was included. Calories vary quite a
bit partly because bars are not of uniform size. Just how many calories should an ice cream bar contain?
342
439
A)
B)
377
111
319
201
353
182
295
197
234
209
294
147
286
190
377
151
Determine a 5-number summary for calories. Check for outliers.
Construct a boxplot for these data. Describe the distribution.
182
131
310
151
Measures of Spread
Range
Variance
Standard Deviation
Variance and Standard Deviation
In the Consumer’s Report April 2007 issue, the following gas mileage was reported in mixed driving
for the following five brands of Subaru:
Subaru B9 Tribeca
16 mpg
Subaru Forester
22 mpg
Subaru Impreza
23 mpg
Subaru Legacy
18 mpg
Subaru Outback
19 mpg
 Find the mean and median.
 Find the variance and standard deviation.
Observations: xi
variance:
standard deviation:
Deviations: xi  x
s2 
b
1
 xi  x
n 1
g
2
b
Squared deviations xi  x
g
2
The following is a list of the number of calories for the 5 top rated brands of hotdogs
(Consumers Report July 2007). Calculate the mean, variance and standard deviation.
150 170 120 120 90
Write the letter of the histogram next to the appropriate variable number in the table below. Explain briefly
how you made your choice.
Variable Mean Median St.Dev.
1
50
50
10
2
50
50
15
3
53
50
10
4
53
50
20
5
47
50
10
6
50
50
5
Consider the hypothetical exam scores presented below for three classes of students. Dotplots of the
distributions are also presented.




Do these dotplots reveal differences among the three distributions of exam scores? Explain briefly.
Calculate the 5-number summaries of the three distributions.
Create the modified boxplots of the three distributions.
If you had not seen the actual data and had only been shown the boxplots, would you have been able to
detect the differences in the three distributions? Describe what feature is difficult to determine from a
boxplot.
Match the following histograms to their corresponding boxplot.
Editors of an Entertainment Weekly publication ranked every episode of Star Trek: The Next Generation from
best (rank 1) to worst (rank 178), as shown in the table, separated according to the season of the show’s sevenyear run in which the episode aired.








Overall, which season was the best? (careful!!!!!) Justify your choice.
Which season was the worst? Justify your choice.
The top 25% of which season was the highest ranked?
Which two seasons seem to have the widest spread?
Which season has the shortest interquartile range?
List the top 3 seasons (from best to worst) based on their third quartiles.
The bottom 50% of which two seasons has practically the same spread?
Which season had the most episodes?
Comparing distributions
Side by side bar graphs
Back to back stemplots
Parallel Boxplots
Teacher salaries in Katy ISD range from $45,000 to $70,000. If the board decides to increase all
salaries by $1,000 for next year, how will that affect the mean and the median? The range and the
standard deviation?
Instead the board decides to go with a 3% increase. How will that affect the mean and median? The
range and standard deviation?
Effects of linear transformations
Adding a constant value
Multiplying by a constant
Maria measures the lengths of 5 cockroaches that she finds at school. Here are her results (in inches):
1.4
2.2
1.1
1.6
1.2
a) Find the mean, median, range and standard deviation of Maria’s measurements
b) Maria’s science teacher is furious to discover that she has measured the cockroach lengths in inches
rather than centimeters. (There are 2.54 cm in 1 inch.) She gives Maria two minutes to report the mean
and standard deviation of the 5 cockroaches in centimeters. Maria succeeded. Will you?