# Frequency distribution table

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This pdf includes the following topics:-
Frequency Distributions
EPAGAS
Frequency Table
Relative Frequency
Choosing Categories
Bar Graphs
Representing Quantitative data using a Histogram and many more.
1. Frequency Distributions
In this section, we look at ways to organize data in order to
make it user friendly. We begin by presenting two data
sets, from which, because of how the data is presented, it is
difficult to obtain meaningful information. We will present
ways to organize and present the data , from which
meaningful summary information can be derived at a
glance.
Data Set 1 A random sample of 20 students were asked
to estimate the average number of hours they spent per
week studying outside of class. Also their eye color and the
number of pets they owned was recorded. The results are
given on the next page.
2. Frequency Distributions
Student # Hours Studying Eye Color # Pets
Student 1 10 blue 1
Student 2 7 brown 0
Student 3 15 brown 3
Student 4 20 green 1
Student 5 40 blue 2
Student 6 25 green 1
Student 7 22 hazel 0
Student 8 13 brown 5
Student 9 12 gray 4
Student 10 21 hazel 3
Student 11 16 blue 1
Student 12 22 green 1
Student 13 25 brown 1
Student 14 30 green 2
Student 15 29 brown 0
Student 16 25 green 4
Student 17 27 gray 0
Student 18 15 hazel 1
Student 19 14 blue 2
Student 20 17 brown 2
3. Frequency Distributions
Data Set 2: EPAGAS The Environmental Protection
Agency (EPA) perform extensive tests on all new car
models to determine their mileage ratings. The 25
measurements given below represent the results of the test
on a sample of size 25 of a new car model.
EPA mileage ratings on 25 cars
36.3 41.0 36.9 37.1 44.9
40.5 36.5 37.6 33.9 40.2
38.5 39.0 35.5 34.8 38.6
41.0 31.8 37.3 33.1 37.0
37.1 40.3 36.7 37.0 33.9
4. Frequency Table or Frequency Distribution
To construct a frequency table, we divide the observations
into classes or categories. The number of observations in
each category is called the frequency of that category. A
Frequency Table or Frequency Distribution is a table
showing the categories next to their frequencies. When
dealing with Quantitative data (data that is numerical in
nature), the categories into which we group the data may
be defined as a range or an interval of numbers, such as
0 − 10 or they may be single outcomes (depending on the
nature of the data). When dealing with Qualitative data
(non-numerical data), the categories may be single
outcomes or groups of outcomes. When grouping the data
in categories, make sure that they are disjoint (to ensure
that observations do not fall into more than category) and
that every observation falls into one of the categories.
5. Frequency Table or Frequency Distribution
Example: Data Set 1 Here are frequency distributions
for the data on eye color and number of pets owned. (Note
that we lose some information from our original data set by
separating the data)
Eye Color # of Students # Pets # of Students
(Category) ( Frequency)
(Category) ( Frequency)
0 4
Blue 4
1 7
Brown 6
2 4
Gray 2
3 2
Hazel 5 4 2
Green 3 5 1
Total 20 Total 20
Note that sum of frequencies = total number of
observations, in this case number of students in our sample.
6. Relative Frequency
The relative frequency of a category is the frequency of
that category (the number of observations that fall into the
category) divided by the total number of observations:
Relative Frequency of Category i =
frequency of category i
total number of observations
We may wish to also/only record the relative frequency
of the classes (or outcomes) in our table.
7. Relative Frequency
Eye Color Proportion of Students # Pets Proportion of Students
(Category) ( Rel. Frequency)
(Category) ( Rel. Frequency)
0 0.20
Blue 0.20
1 0.35
Brown 0.30
2 0.20
Gray 0.10 3 0.10
Hazel 0.25 4 0.10
Green 0.15 5 0.05
Total 1.0
Total 1.0
8. Choosing Categories
I When choosing categories, the categories should
cover the entire range of observations, but
should not overlap. If the categories chosen are
intervals one should specify what happens to data at
the end points of the intervals.
I For example if the categories are the intervals 0-10,
10-20, 20-30, 30-40, 40-50. One should specify which
interval 10 goes into, which interval 20 goes into, etc..
It’s usual to use different brackets in interval notation
to indicate whether the endpoint is included or not.
The notation [0, 10) denotes the interval from 0 to 10
where 0 is included in the interval but 10 is not.
9. Choosing Categories
I Common sense should be used in forming categories.
Somewhere between 5 and 15 categories gives a
meaningful picture that is easily processed.
However if there are only 3 candidates for a
presidential election and you conduct a poll to
determine who those polled will vote for, then it is
natural to choose 3 categories.
10. Choosing Categories
I Common sense should be used in forming categories.
Somewhere between 5 and 15 categories gives a
meaningful picture that is easily processed.
However if there are only 3 candidates for a
presidential election and you conduct a poll to
determine who those polled will vote for, then it is
natural to choose 3 categories.
I To choose intervals as categories with quantitative
data, one might subtract the smallest observation from
the largest and divide by the desired number of
intervals. This gives a rough idea of interval length.
Then adjust it to a simpler (larger) number which is
relatively close to it, making intervals of the desired
length where the first starts at a natural point lower
than the minimum observation and the last ends at a
natural point greater than the maximum observation.
11. Choosing Categories
I For example, if you data ranged from 1 to 29, and you
wanted to create 6 categories as intervals of equal
length. The length of each should be approximately
29−1
6 ≈ 4.667. It is natural to use 6 intervals of length
5 in this case, with the first starting at 0 and the last
ending at 30. If we decide to include the right end
point and exclude the left end point for each interval,
our intervals are :
(0, 5], (5, 10], (10, 15], (15, 20], (20, 25], (25, 30].
12. Choosing Categories
Example: Data set 2 Make a frequency distribution
(table) for the data on mileage ratings using 5 intervals of
equal length. Include the left end point of each interval and
omit the right end point.
EPA mileage ratings on 25 cars
Mileage # of cars 36.3 41.0 36.9 37.1 44.9
(Category) ( Frequency) 40.5 36.5 37.6 33.9 40.2
38.5 39.0 35.5 34.8 38.6
[ , ) 41.0 31.8 37.3 33.1 37.0
37.1 40.3 36.7 37.0 33.9
[ , )
[ , )
[ , )
[ , )
Total
13. Choosing Categories
We are told to divide the data into 5 intervals of equal length.
The smallest value in the data is 31.8 and the largest is 44.9
44.9 − 31.8
and = 2.62. If we start at 30.0 and use intervals of
5
length 3, 5 intervals later will end at 45.0 so we cover the data.
Mileage # of cars
(Category) ( Frequency)
[30, 33 ) 1
[33, 36) 5
The value 39.0 goes in the in-
[36, 39 ) 12 terval [39, 42) NOT the inter-
val [36, 39).
[39, 42 ) 6
[42, 45 ) 1
Total 25
14. Choosing Categories
Example: Data set 1 Make a frequency distribution
(table) for the data on the estimated average number of
hours spent studying in data set 1, using 7 intervals of
equal length. Include the left end point of each interval and
omit the right end point.
We are told to divide the data into 7 intervals of equal
length. The smallest value in the data is 7 and the largest
40 − 7
is 40. Since ≈ 4.7, it makes sense to use intervals of
7
length 5. Starting at 5, we will end at 40. Since we have a
value of 40 and we have agreed to omit right-hand end
points, this does not quite work. If we start with 6 we will
be OK.
15. Choosing Categories
Hours Studying # of students
(Category) ( Frequency)
[6, 11 ) 2
[11, 16 ) 5
[16, 21 ) 3
[21, 26 ) 6
[26, 31 ) 3
[31, 36 ) 0
[36, 41 ) 1
Total 20
16. If we started with 5 and used 8 intervals:
Hours Studying # of students
(Category) ( Frequency)
[5, 10 ) 1
[10, 15 ) 4
[15, 20 ) 4
[20, 25 ) 4
[25, 30 ) 5
[30, 35 ) 1
[35, 40 ) 0
[40, 45 ) 1
Total 20
17. Representing Qualitative data graphically
Pie Chart One way to present our qualitative data
graphically is using a Pie Chart. The pie is represented by
a circle (Spanning 3600 ). The size of the pie slice
representing each category is proportional to the relative
frequency of the category. The angle that the slice makes
at the center is also proportional to the relative frequency
of the category; in fact the angle for a given category is
given by:
category angle at the center =
relative frequency category × 3600 .
The pie chart should always adhere to the area principle.
That is the proportion of the area of the pie devoted to any
category is the same as the proportion of the data that lies
in that category. This principle is commonly violated to
alter perception and subtly promote a particular point of
view (see end of slides).
18. Representing Qualitative data graphically
Example 1 Here is the data on eye color from data set 1
in a pie chart.
19. My favourite pie chart
20. Bar Graphs
We can also represent our data graphically on a Bar
Chart or Bar Graph. Here the categories of the
qualitative variable are represented by bars, where the
height of each bar is either the category frequency, category
relative frequency, or category percentage.
The bases of all bars should be equal in width. Having
equal bases ensures that the bar graph adheres to the area
principle, which in this case means that the proportion of
the total area of the bars devoted to a category( = area of
the bar above a category divided by the sum of the areas of
all bars) should be the same as the proportion of the data
in the category. This principle is often violated to promote
a particular point of view (see end of slides).
21. Bar Graphs
22. Representing Quantitative data using a Histogram
Histograms A histogram is a bar chart in which each
bar represents a category and its height represents either
the frequency, relative frequency (proportion) or percentage
in that category.
If a variable can only take on a finite number of values (or
the values can be listed in an infinite sequence) the variable
is said to be discrete.
For example the number of pets in Data set 1 was a
discrete variable and each value formed a category of its
own. In this case, each bar in the histogram is centered
over the number corresponding to the category and all bars
have equal width of 1 unit. (see below).
23. Representing Quantitative data using a Histogram
24. Representing Quantitative data using a Histogram
If a variable can take all values in some interval, it is called
a continuous variable. If our data consists of observations
of a continuous variable, such as that in data set 2, the
categories used for our histogram should be intervals of
equal length (to adhere to the area principle) formed in a
manner similar to that described above for frequency
tables. The bases of the bars in our histogram are
comprised of these categories of equal length and their
heights represent either the frequency, relative frequency or
percentage in each category. Because it is difficult to tell
from the histogram alone which endpoints are included in
the categories, we adopt the convention that the categories
(intervals) include the left endpoint but not the right
endpoint.
25. Representing Quantitative data using a Histogram
Example Construct a histogram for the data in data set 2
on EPA mileage ratings, using the categories used above in
the frequency table. Use the frequency of observations in
each category to define the height of the bars.
Mileage # of cars
(Category) ( Frequency)
[ , )
[ , )
[ , )
[ , )
[ , )
Total
26. Representing Quantitative data using a Histogram
On the left is the frequency data from above.
Hours Studying # of students
(Category) ( Frequency)
[6, 11 ) 2
[11, 16 ) 5
[16, 21 ) 3
[21, 26 ) 6
[26, 31 ) 3
[31, 36 ) 0
[36, 41 ) 1
Total 20
27. Changing the width of the categories
For large data sets one can get a finer description of the
data, by decreasing the width of the class intervals on the
histogram. The following Histograms are for the same set
of data, recording the duration (in minutes) of eruptions of
the Old Faithful Geyser in Yellowstone National Park.
01/07/2008 06:34 PM
28. Stem and Leaf Display
Another graphical display presenting a compact picture of
the data is given by a stem and leaf plot.
To construct a Stem and Leaf plot
I Separate each measurement into a stem and a leaf –
generally the leaf consists of exactly one digit (the last
one) and the stem consists of 1 or more digits.
e.g.: 734 stem = 73, leaf=4
2.345 stem = 2.34, leaf=5.
Sometimes the decimal is left out of the stem but a note is
added on how to read each value. For the 2.345
example we would state that 234|5 should be read as 2.345.
29. Stem and Leaf Display
Sometimes, when the observed values have many
digits, it may be helpful either to round the numbers
(round 2.345 to 2.35, with stem=2.3, leaf=5) or truncate
(or dropping) digits (truncate 2.345 to 2.34).
I Write out the stems in order increasing vertically (from
top to bottom) and draw a line to the right of the
stems.
I Attach each leaf to the appropriate stem.
I Arrange the leaves in increasing order (from left to
right).
30. Stem and Leaf Display
Example Make a Stem and Leaf Plot for the data on the
average number of hours spent studying per week given in
Data Set 1.
10, 7, 15, 20, 40, 25, 22, 13, 12, 21
16, 22, 25, 30, 29, 25, 27, 15, 14, 17
All are data points are 2 digit integers and the tens digit
goes from 0 to 4.
0 7
1 0 2 3 4 5 5 6 7
2 0 2 3 4 5 5 6 7 9
3 0
4 0
31. Extras : How to Lie with statistics
Example This (faux) pie chart, shows the needs of a cat,
and comes from a box containing a cat toy. Note that the
“categories” are not distinct and they use an exploding
slice to distort the are for Hunting, which is the need of
your cat that this particular toy is supposed to fulfill.
32. Extras : How to Lie with statistics
33. Extras : How to Lie with statistics
A subtle way to lie with statistics is to violate the area
rule. The pie chart below is distorted to make the areas of
regions devoted to some categories proportionally larger
than they should be by stretching
73492685_3d516242aa_m.jpg the pie
(JPEG Image, intopixels)
240x198 an oval
shape and adding a third dimension.
34. Extras : How to Lie with statistics
Example Both of the following graphs represent the same
information. The graph on the left violates the area
principle by making the base of the bars (banknotes) of
unequal width.
Purchasing Power of the Diminishing Dollar
\$1.00
1.0
94c
83c
0.8
64c
0.6
44c
0.4
0.2
0.0
1958 1963 1968 1973 1978
Eisenhower Kennedy Johnson Nixon Carter
Is the bottom dollar note roughly half the size of the top one?
35. Google Image Result for http://lilt.ilstu.edu/gmklass/pos138/datadisplay/sections/charts/3%20graphic%20data_files/image016.jpg 02/10/2007 07:43 PM
Extras : How to Lie with statistics Google Image Result for http://lilt.ilstu.edu/gmklass/pos138/datadis... http://images.google.com/
a number (actually, in the case of a scatterplot, two numbers). It is the job of the chart’s text to
tell the reader just what each of those numbers represents.
See full-size image.
Designing good charts, however, presents more challenges than tabular display as it draws on
lilt.ilstu.edu/.../image016.jpg
Example
the talents of both All ofandthe
the scientist the artist.following
You have to know andgraphs
understand your violate
data, but the area 504 x 389 pixels - 27k
you also need a good sense of how the reader will visualize the chart’s graphical elements. Image may be scaled down and su
principle by replacing the bars displayed
Two problems arise in charting that are less common when data areBelow
by irregular
in tables. Poor
objects in
Example 4.4 How to Lie with Statistics is the image in its original context on the page: lilt.ilstu.edu/.../section
addition to making
choices, or deliberately deceptive, choicesthe bases
in graphic of unequal
design can provide a distorted picture oflength.
The bar graph that follows presents the total sales figures for three realtors.
When the bars are replaced with pictures, often related to the topic of the
numbers
graph, the graph is called and relationships they represent.
a pictogram. A more common problem is that charts are often
Total designed in ways that hide what the data might tell us, or that distract the reader from quickly
Sales \$2.05 million
discerning the meaning of the evidence presented in the chart. Each of these problems is
\$1.41 million
illustrated in the two classic texts on data presentation: Darrell Huff’s How to Lie with Statistics
\$0.9 million
(1994) and Edward Tufte’s The Visual Display of Quantitative Information (1983).
Huff’s
No. #1
Realtor 1 little paperback,
No. 2#2 first published
Realtor RealtorNo.
#33 in 1954 and reissued many times thereafter, condemned
Realtor
graphical representations of data that “lied”. Here, the two numbers, one 3 times the magnitude
(a) How does the height of the home for Realtor 1 compare to that for
of3?the other, are represented by two cows, one
Realtor 27 times larger than the other, resulting in a Lie
(b) How does the area of the home for Realtor 1 compare to that for
Factor
Realtor 3? of 9.
Solution
(a) The height for Realtor 1 is just slightly over twice that of Realtor 3. The
heights are at the correct total sales levels.
(b) To avoid distortion of the pictures, the area of the home for Realtor 1 is
more than four times the area of the home for Realtor 3.
What We’ve Learned: When you see a pictogram, be careful to interpret the
results appropriately, and do not allow the area of the pictures to mislead you.
!
Figure 1: Graphical distortion of data
SOURCE: DarrellChapter
Huff.4 --- 13 1993. How to Lie
with Statistics WW Norton & Co, 72.
Here the figure depicts the increase in the number of milk cows in the United States, from 8
million in 1860 to twenty five million in 1936. The larger cow is thus represented as three
times the height the 1860 cow. But she is also three times as wide, thus taking up nine times the