Unit 5 Descriptive Statistics Day 2
Data: It’s Not Just for Math Class Anymore
Name: _________________________
Objectives:
S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread
(inter-quartile range, standard deviation) of two or more different sets.
S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible
effects of extreme data points (outliers).
Part I: DESCRIBING SHAPE OF A DATA DISTRIBUTION
When describing a distribution, it is important to include information about its shape, its center, and its spread.
Describing shape. Some distributions are approximately normal or mound shaped, where the distribution has one
peak and tapers off on both sides. Normal distributions are symmetric, the two halves look like mirror images of
each other. Some distributions have a tail stretching towards the larger values. These distributions are called
skewed to the right or skewed toward the larger values. Distributions that have a tail stretching toward the
smaller values are called skewed to the left or skewed toward the small values.
Uniform
“symmetric”
*A description of shape should include whether there are two or more clusters separated by gaps.
*You should also identify any outliers (unusually large or small values that fall outside the overall pattern)
Example #1: As part of an effort to study the wild black bear population in Minnesota, Department of Natural
Resources staff anesthetized and then measure the length of 143 black bears. (The length of a bear is measured
from the tip of its nose to the tip of its tail.) The following dot plots show the distributions of the length of the male
and the female bears.
Use the ideas of skewness and outliers to describe the shape of the distribution of lengths of female black bears.
Be specific!
1
Example #2: The histograms below show the heights (rounded to the nearest inch) of large samples of young adult
men and women in the United States.
a. Describe the shapes of these two distributions. Be specific (skew, outliers, clusters/gaps)
b. How do the heights of men compare to women. Use specific data and the graphs to back up your
statements.
c. Which distribution has the larger spread (range) in the values? Explain how you know/show your work.
Heights of Young Adult Women
Frequency
Frequency
Heights of Young Adult Men
d. How many total men were surveyed? ____________ How do you know?
e. How many total women were surveyed?______________ How do you know?
f.
How many men were exactly 6 feet tall?
g. How many women were at least 5’9?”
**What percent is this? _________
**What percent is this? _________
h. Walt Disney World recently advertised for singers to perform in Beauty and the Beast – live on Stage.
Two positions were Belle, with height 5’5” – 5’8”, and Gaston, with height of 6’1” or taller. How many of
these women would meet the requirement? How many of these men would meet the requirement?
2
Part II: The Effect of Outliers on Measures of Center
#1) Calculate the mean and median of the following set of values:
1, 2, 3, 4, 5, 6, 70
a. Remove the outlier of 70. Then find the mean and median of the new set of values. Explain which
measurement changed more and why this was the case.
b. As a group, create another set of values with a LOW outlier. Find the mean and median. Then remove the
outlier and find the mean and median of the new set of values? Which change more in these cases?
c. In general, is the mean or median more affected by outliers? Explain why this is. How does this change
depending on if it was a high outlier or low outlier?
#2) Your teacher grades on a scale of 0 to 4. Your homework grades are shown in the table.
a) When computing your final grade in the course, would you rather have your teacher use
the mean or median? Justify your answers using mathematical evidence and specific
calculations.
b) Suppose your teacher forgot to record one of your grades in the table. After it is added
to the table, your new mean is 2.5. What was the missing grade? Show your work!
#3) The average height of 10th grade students at Hickman and Rock Bridge are shown below.
Hickman 10th Graders’ Heights
a)
Rock Bridge 10th Graders’ Heights
Describe the shape of the distribution of height for sophomores at both schools. Be specific.
b) How could these histograms be potentially misleading to a person comparing the two groups?
3
Part III: Analyzing Various Representations
1. Sit-Up Contest for One Minute
Type of graph: ______________
a) Describe the distribution for the Sit-Up Contest for One Minute using the terms from above.
b) Find the mean, median, mode, and range of the data if possible. If it is not possible, explain why.
c) This situation has an outlier. It is ______. Recalculate all measurements from (b) WITHOUT this value.
d) Describe how the outlier affected each of the data measurements? If there is no change, explain why.
e) What would you say is the most representative “average” for this data? Explain why.
f)
What conclusion(s) can you could make about Sit-Up Contest for One Minute from this representation?
4
2. Greeting Card Prices
Type of graph: ________________
a)
Describe the distribution for the
Greeting Card Prices using the terms
from above.
b) Find the mean, median, mode, and range of the data if possible. If it is not possible, explain why.
c) What real-life conclusion(s) can you could make about Greeting Card Prices from this representation?
3.
Classmates’ Oldest Sibling’s Age
Type of graph: ________________
a)
Describe the distribution for the Classmates’ Oldest
Sibling’s Age using the terms from above.
b) Find the mean, median, mode, and range of the data if possible. If it is not possible, explain why.
c) What real-life conclusion(s) can you could make about Classmates’ Oldest Sibling’s Age from this
representation?
d)
If a new student joined the class and had a sibling who was 58, how would this affect our data’s measures
of center? Be specific, use calculations to justify your answers.
5