Math 3
Investigation: Pencil Length
April 16, 2012
MM3D1. Students will create probability histograms of discrete random variables, using both experimental and
theoretical probabilities.
In this investigation you’ll explore the differences between discrete and continuous random variables.
16
16.3
16.3
13.6
17.5
17.7
17
12.5
17.2
13.1
17.4
14.9
15.8
17.8
14.6
10.3
14.5
18
13.9
11.2
15.05
15.5
12.2
14.1
16.1
17
18.3
17.5
Step 1
Using the pencil length data we collected Friday (shown above), create a histogram on the
graph paper provided with classes representing 1 cm increments in pencil length.
Step 2
Divide the number of pencils in each bin (class) by the total number of pencils. Make a new
histogram using these quotients as the values on the y-axis.
Step 3
Check that the area of your second histogram is 1, why must this be true?
Step 4
Imagine that you collect more and more pencils and draw a histogram using the method
described in Step 3. Sketch what this histogram of increasingly many pencil lengths would
look like. Give reasons for your answer.
Step 5
Imagine doing a very complete and precise survey of all the pencils in the world. Assume that
their distribution is about the same as the distribution of pencils in your sample. Also assume
that you use infinitely many very narrow bins.
What will this histogram look like?
To approximate this plot, sketch over the top of
your histogram with a smooth curve, as shown
to the right. Make the area between the curve
and the horizontal axis about the same as the
area of the histogram. Make sure that the extra
area enclosed by the curve above the histogram
is about the same as the area cut off the corners
of the bins as you smooth out the shape.
Step 6
Let x represent pencil length. Use your histogram from Step 5 to estimate the areas of various
regions between the curve and the x-axis that satisfy these conditions:
a. x  10
b. 11  x  12
c. x  12.5
d. x  11
The histogram you made in Step 2 of the investigation, giving the proportions of pencils in the bins, is a
relative frequency histogram. It shows what fraction of the time the value of a discrete random variable falls
within each bin. The continuous curve you drew in Step 5 approximates a continuous random variable for the
infinite set of measurements. This graph represents a function called a probability distribution.
The areas you found in step 6 of the investigation give probabilities that a randomly chosen pencil length will
satisfy a condition. These probabilities are given by areas. If x represents the continuous random variable giving
the pencil lengths in centimeters, then you can write these areas as
P(x < 10 cm), P(11 cm < x < 12 cm), P (x > 12.5 cm) , and P(x = 11 cm)
In a continuous probability distribution, the probability of any single outcome, such as the probability that x is
exactly 11 centimeters, is the area of a line segment, which is 0. It is possible for a pencil to be exactly 11 cm
long, but the probability of choosing any one value from infinitely many values is theoretically 0. As you
learned have learned, a single point or line has no area.
In the following example, you’ll see how areas represent probabilities for a continuous random variable.