Transcript SWBAT: Measure standard deviation, identify outliers, and construct

SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
Do Now: Explain how this graph
could be misleading.
SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
Do Now: The total return on a stock is the change in its market price plus any
dividend payments made. Total return is usually expressed as a percent of the
beginning price. The histogram below shows the distribution of total returns for all
1528 stocks listed on the New York State Exchange in one year. This is a histogram of
the percents in each class rather than a histogram of counts.
(a) Describe the overall shape of the
distribution of total returns.
(b) What is the approximate center of this
distribution?
(c) Approximately what were the smallest and largest total returns? (Describe the
spread of the distribution)
SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
The Interquartile Range (IQR)
To calculate the quartiles:
Arrange the observations in increasing order and locate
the median M.
The first quartile Q1 is the median of the observations
located to the left of the median in the ordered list.
The third quartile Q3 is the median of the observations
located to the right of the median in the ordered list.
The interquartile range (IQR) is defined as:
IQR = Q3 – Q1
(this measures the range of the middle 50%)
SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
Example: Find and Interpret the IQR for the travel
times to work for 20 randomly selected New Yorkers.
SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
Use the IQR rule to identify outliers:
1.5 x IQR
SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
The five-number summary:
The five-number summary of a distribution consists of the smallest
observation, the first quartile, the median, the third quartile, and the
largest observation, written in order from smallest to largest. In symbols,
the five-number summary is
Minimum Q1 M Q3 Maximum
This five number summary leads to a new graph displayed as
Boxplot: A graph of the five-number summary.
SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
Example 1: The 2009 roster of the Dallas Cowboys professional football team
included 10 offensive linemen. Their weights (in pounds) were
338 318 353 313 318 326 307 317 311 311
a) Find the five-number summary for these data by hand. Show your work.
b) Calculate the IQR. Interpret this value in context.
c) Determine whether there are any outliers using the 1.5 × IQR rule.
d) Draw a boxplot of the data.
SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
Example 2: Barry Bonds set the major league record by hitting 73 home runs in
a single season in 2001. On August 7, 2007, Bonds hit his 756th career home run,
which broke Hank Aaron’s longstanding record of 755. By the end of the 2007
season when Bonds retired, he had increased the total to 762. Here are data
on the number of home runs that Bonds hit in each of his 21 complete seasons:
16, 25, 24, 19, 33, 25, 34, 46, 37, 33, 42, 40, 37, 34, 49, 73, 46, 45, 45, 26, 28
a) Find the five-number summary for these data by hand. Show your work.
b) Calculate the IQR. Interpret this value in context.
c) Determine whether there are any outliers using the 1.5 × IQR rule.
d) Draw a boxplot of the data.
SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
Do Now:
a) Determine the five number summary for the data.
b) Find the IQR.
c) Are there any outliers?
d) Create a boxplot of your data.
SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
Standard Deviation: sx measures the average distance of the observations from
their mean.
Example: Let’s examine data on the number of pets owned by a group of 9 children.
Here are the data, arranged from lowest to highest:
1 3 4 4 4 5 7 8 9
Variance: The average squared distance of the
observations in a data set from their mean.
SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
Example 1: The heights (in inches)of the five starters on a basketball team are
67, 72, 76, 76, and 84.
(a) Find and interpret the mean.
(b) Make a table that shows, for each value, its deviation from the mean and its
squared deviation from the mean.
(c) Show how to calculate the variance and standard deviation from the values in your
table. Interpret the meaning of the standard deviation in this setting.
SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
Example 2: The first four students to arrive for a first-period statistics class were
asked how much sleep (to the nearest hour) they got last night.
Their responses were 7, 7, 9, and 9.
(a) Find the standard deviation from its definition.
(b) Interpret the value of sx you obtained in (a).
(c) Do you think it’s safe to conclude that the mean amount of sleep for all 30
students in this class is close to 8 hours? Why or why not?
SWBAT: Measure standard deviation, identify outliers, and
construct a boxplot using the five-number summary
*The median and IQR are usually better than the mean and
standard deviation for describing a skewed distribution or a
distribution with strong outliers.
*Use x and sx only for reasonably symmetric distributions that
don’t have outliers.