2.5 Histograms
Histograms with evenly-sized bins
A frequency distribution is a table that displays how often an outcome occurs
for a sample. To construct a frequency distribution, the data set is divided into
mutually exclusive
classes
. A class is either a value of a categorical variable
or an interval of a continuous variable. The frequency of a class is the number
of events or values that fall under each class. Ex: An informal poll among a
group of friends tallies how many people have
x
gaming applications on their
phone. The results of the poll can be summarized in the frequency distribution
below.
Table 2.5.1: Frequency distribution showing the number of people in a
group having x gaming apps on their phone.
Gaming apps (x)
Tally
Frequency
0
||||
4
1
||||
||
7
2
||||
5
3
|||
3
4
||
2
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The most common graphical representation of a frequency distribution is a
histogram. A histogram depicts data values by splitting a continuous variable
into a number of class intervals, each known as a bin. The simplest and most
common type of histogram has bins of equal size. When bin sizes are equal,
bins have rectangular bars with heights representing the frequency, which is
the number of values in a bin.
The
x
-axis contains a continuous number line with ticks that represent bin
boundaries. A bin includes values equal to or greater than the lower boundary,
but less than the upper boundary (lower
≤
value
<
upper). Gaps between
rectangles are removed to show that the data is continuous.
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2.5.1: Histogram of number of tickets per miles per hour over speed limit.
1234

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2x speed
24
16
17
12
10
14
19
4
13
18
9
11
9
24
28
24
14
16
12
10
19
4
24
18
11
13
9
9
28
0
5
10
15
20
25
30
MPH over speed limit for 15 tickets issued:
2
4
6
Frequency
(number of tickets)
MPH over speed limit
17
1
3
5
Each bin frequency is represented by a rectangular column.
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2.5.2: Histogram fundamentals.

Consider the following histogram showing speeding tickets issued by Officer
Brown:
1)
3
speeding tickets were issued between
5−10
mph.
True
False
Correct
The rectangle for the 5−10 mph bin has a height of 3.
2)
The most speeding tickets were given in the
15−20
mph range.
True
False
Correct
The rectangle for the 15−20 mph, at height 5, is taller than any other bin's rectangle.
3)
The number of speeding tickets issued between
0−5
mph is unknown.
True
False
Correct
An empty bin indicates the frequency is (known to be) 0, so no tickets were issued
between 0−5 mph.
4)
A ticket issued that is
5
mph over the speed limit should be placed in
the
0−5
mph bin.

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True
False
Correct
A 5 mph ticket should be placed in the 5−10 mph bin because a number on the upper boundary of a
bin's range (ex: 0−5 mph bin) should be placed in the next bin.

- Fall '18