Chapter 7
Exploring Measures of Variability
Objectives
Students will be able to:
1) Calculate mean absolute deviation and
standard deviation, and use these values to
measure consistency
2) Test for a difference in standard deviations
• In sports, what does it mean to be consistent, and
why does consistency matter?
• Here are the 2013 passer ratings for Eli Manning:
102.3
53.3
49.0
64.8
56.1
58.5
81.1
81.8
70.3
92.4
92.9
98.7
72.3
31.9
71.1
Would you say Manning’s PERFORMANCES were
consistent?
• To be consistent means that an athlete’s or
team’s PERFORMANCES are very similar to
each other.
• Examples:
– In basketball, a consistent player will score about
the same number of points each game.
– In swimming, a consistent swimmer will swim
about the same time in each race.
– In football, a consistent running back will gain
about the same number of yards each game.
• Here are the distributions for two golfers. Both average
200 yards per drive. Which is more consistent? How can
we measure the variability of each distribution?
– Range
-IQR
• What are some problems with these measures?
– The range is influenced by outliers.
– IQR only measures the spread of the middle half of
the observations, so it doesn’t tell us about the
variability of the entire distribution.
• How can we measure variability using each
value in a distribution?
– Measure how far each value is from the center of
the distribution, then find an average distance to
the center.
• This chapter we will focus on two measures of
spread that use every PERFORMANCE in a
distribution: the mean absolute deviation and
the standard deviation.
• These differ from range and IQR which only
measure the distance between two positions
in a distribution.
Mean Absolute Deviation (MAD)
• How far is each of these points from the
mean?
• A deviation measures the distance between
an observed PERFORMANCE and the mean of
its distribution:
Deviation = PERFORMANCE – mean
• The mean absolute deviation (MAD) measures
the average distance the values in a
distribution are from their mean.
• Former Chicago White Sox manager Ozzie Guillen
once complained about his team’s lack of
consistency.
• From April 24 to May 4, 2010 (10 games), the
White Sox opponents scored the following run
totals:
4 2 4 6 5 6 6 12 1 7
• To get an overall measure of how variable these
PERFORMANCES were from the mean, let’s
calculate the MAD for this distribution.
Steps to Calculate MAD
1) Calculate the mean PERFORMANCE.
4 2 4 6 5 6 6 12 1 7
2) Calculate the deviations from the mean
PERFORMANCE.
actual PERFORMANCE – mean PERFORMANCE
• If a PERFORMANCE is above average, the
deviation will be positive.
• If a PERFORMANCE is below average, the
deviation will be negative.
• The chart is also on pg 227.
3) Find the absolute value of each deviation.
Why would we want to do this?
If we were to simply add the deviations, the sum
would be 0.
4) Calculate the mean of the absolute deviations.
On average, the number of runs allowed by the
White Sox was 2.1 runs from their mean.
Let’s compare this to the number of runs the White
Sox scored themselves over these 10 games.
• Which distribution of PERFORMANCES looks
more consistent?
• It looks like the runs scored is more consistent.
To confirm, let’s compare the MAD for each
distribution.
Calculate the MAD for the runs scored
distribution:
5 3 2 5 7 4 7 3 5 2
• The MAD for the runs allowed distribution is
2.1 runs and the MAD for the runs scored
distribution is 1.5 runs. Is our hunch
confirmed?
• The runs scored distribution is more
consistent.
• The smaller the MAD, the more consistent the
PERFORMANCES are.
The Standard Deviation
• When calculating mean absolute deviation,
we had to use absolute value to assure that
each deviation would be positive.
• Can anyone think of another way we could
have made each deviation positive?
– Squaring each deviation
• Standard deviation measures the variability in a
distribution using the squared deviations from the mean.
• You may be asking yourself “What are the benefits to
using standard deviation over MAD?”
– 1) We will be working a lot with the Normal distribution
(spoiler alert: next chapter!) and the Normal
distribution is defined in terms of the standard
deviation
– 2) Many important techniques in statistics are based on
the idea of squared deviations, such as least-squares
regression lines (Chapter 11)
What does standard deviation measure?
• Standard deviation measures the typical
distance between an athlete’s
PERFORMANCES and his or her ABILITY. In
other terms, it is the typical distance between
observations and the mean.
• To better understand this, let’s look at an
example…
Here are 82 simulated
basketball games for 3
different players. (pg 231)
-The first player has an
ABILITY to score 20 points
per game, and has a
standard deviation of 5
points per game.
-Second player:
-20 points per game
-Standard deviation of 10
-Third player:
-20 points per game
-standard deviation of 2
• The first player’s
average is 20 points
per game. However,
the individual game
PERFORMANCES
varied somewhat, due
to RANDOM CHANCE.
• The standard deviation is 5, meaning typically his
PERFORMANCES were about 5 points from his ABILITY.
• Player 2 had a
standard deviation
of 10, meaning
typically his
PERFORMANCES
were about 10
points from his
ABILITY.
• Player 3’s
PERFORMANCES
were typically
about 2 points
from his ABILITY.
• Which player
performs more
consistently?
– Player 3
• As with MAD, the
smaller the standard
deviation, the more
consistent the
PERFORMANCES.
Calculating the Standard Deviation
• One thing to keep in mind is that standard
deviation will be a little larger than the mean
absolute deviation. Why might this be?
• Instead of taking absolute value of the
deviations, we are squaring them. This will
give extra weight to values far from the mean.
• Let’s look at the steps to calculate standard
deviation.
• We will use our previous runs allowed data
from the White Sox example.
4 2 4 6 5 6 6 12 1 7
• The first two steps are exactly the same as the
first to steps to calculating the MAD.
• Step 1: Calculate the mean.
• Step 2: Find the deviations from the mean.
(pg 234)
• Step 3: Square each deviation.
• Step 4:
a) Add the standard deviations
b) Divide this total by 1 less than the total number
of observations (n-1)
c) Take the square root.
Using Technology to Calculate the
Standard Deviation
Notation
• The standard deviation of a set of
PERFORMANCES is denoted by a lowercase
“s.”
• On the TI-84, we can find the standard
deviation the same way we previously found
summary statistics.
Steps to Calculate Standard Deviation on the TI-84
Let’s use the same 10 White Sox observations:
4 2 4 6 5 6 6 12 1 7
1) Enter the observations into a list.
2) Press STAT, go to the CALC column, choose 1-Var
Stats, and select your list.
3) The standard deviation is labeled sx. The value
should be 3.02 to match our previous calculation.
• On the iPad, the BStatisticsLite app calculates
standard deviation.
• Let’s try it!!!
Testing for a Difference in Standard Deviations
• Previously, we compared an athlete’s
PERFORMANCES in two different contexts to
investigate if the athlete had a greater ABILITY
in one of those contexts.
• Since we never truly know an athlete’s
ABILITY, we had to estimate it with
PERFORMANCES, which are partly due to
RANDOM CHANCE.
• The same concept is applicable to standard
deviation.
• An athlete’s true standard deviation would be
their standard deviation after an infinite
number of PERFORMANCES.
• Observed standard deviation is standard
deviation based on observed PERFORMANCES.
• Observed standard deviation is used to estimate
true standard deviation. Keep in mind observed
standard deviation will vary from true standard
deviation due to RANDOM CHANCE.
Experiment: Which 7-iron is more
consistent?
• Consistency is very important in golf.
• Knowing exactly how far shots will travel with
each different golf club is a huge strategic
advantage.
• Let’s analyze an experiment to determine if a
golfer is more consistent with a new 7-iron.
• Jimmy is considering buying a new 7-iron,
hoping it will make his shots more consistent.
• To investigate, he decided to conduct an
experiment.
• Luckily for Jimmy, he has a twin brother Sean
that happens to have the new 7-iron that
Jimmy wants to buy.
• Jimmy will use his current 7-iron and borrow
his brother’s 7-iron to see which of the two
clubs makes the distance he hits a 7-iron less
variable.
• What will be the explanatory and response
variables for this experiment?
Explanatory: the club (current or new)
Response: distance the ball travels
• What are some variables we would want to
control?
Same shoes, same type of golf ball, same
gloves, same location, same time, etc…
• Jimmy will hit 20 golf balls in total. How can
randomization be incorporated into the
experiment?
Randomize the order in which the clubs are
used.
Take 20 note cards, write a “C” on 10 for the
current club and a “N” on 10 for the new club.
Shuffle the cards, take one at a time and use
that club.
• Is it possible for Jimmy to be blind?
No. He needs to know what club to use, and
it would be rather difficult to disguise the
clubs.
• Now let’s look at the results…
• Which club looks more consistent?
-new
• Let’s perform a hypothesis test using the
difference in observed standard deviations
(current – new) as the test statistic.
• What are the hypotheses we are interested in
testing?
• The standard deviation for the current 7-iron
is 13.56 yards and the standard deviation of
the new 7-iron is 7.72 yards.
• What is the value of the test statistic?
(current – new)= 13.56-7.72 = 5.84 yards
• Here are 100 trials of this simulation. What is
the p-value?
• 3%
• Conclusion: