a Where is the peak of the graph? How do the bars on the left of the peak look in relation to the bars on the right?
B
b Draw a curve that connects the top of the bars on the histogram.
C
c Most metrics about human physiological characteristics are symmetrically distributed.
A
a See solution.
B
b Because the curve that connects the tops of the bars looks like a bell.
C
c See solution.
Practice makes perfect
a Let's begin by looking at the histogram from Exploration 1. It shows the chest sizes, measured in inches, of 5738 men in the Scottish Militia.
In Exploration 2, we have the distributions of the heights of 250 adult American males and 250 adult American females.
These three distributions are roughly symmetric. This type of distribution has a peak in the center that divides it into two equal halves. The left half is a mirror image of the right half.
Most of the data is located in the center of the distribution. An ideal symmetric distribution has only one mode. Also, the mean, median, and mode are approximately the same.
b When we look at a symmetric distribution, we can connect the top of the bars with a smooth curve.
Since the curve has a similar shape to that of a bell, symmetric distributions are commonly called bell-shaped.
c As our first real-life scenario of a symmetric distribution, let's consider rolling two fair dice and finding their sum. We can create a table of the possible outcomes and their probabilities.
Outcome X
Probability of X, P(X)
2
1/36≈ 0.027
3
2/36≈ 0.055
4
3/36≈0.083
5
4/36≈0.111
6
5/36≈0.138
7
6/36≈0.166
8
5/36≈0.138
9
4/36≈0.111
10
3/36≈0.083
11
2/36≈0.055
12
1/36≈0.27
We can now graph the data. The vertical label will be the probability and the horizontal label will be the possible outcomes. Let's do it!
We can see that the distribution of finding the sum of the dice is symmetric. Another example of a symmetric distribution is the male shoe sizes in the United States. The data has a mean of 10 and a standard deviation of 1. Let's look at its histogram.
Please note that these are two of the many possible examples of real-life symmetric distributions.
