Transcript Document

Chapter 1.2: Describing Distributions with Numbers
Who is a better home run hitter, McGwire, Sosa, or Bonds?
1986
16
1987
25
1988
24
1989
19
1990
33
1991
25
1992
34
1993
46
1994
37
1995
33
1996
42
1997
40
1998
37
1999
34
2000
49
2001
73
Create a stemplot of Bonds record.
Describe the following
1. Center
2. Shape
3. Spread
4. Outliers
Measuring center: the mean
Def) Mean – add values of observations and divide by the
number of observations
1 n
x1  x2  x3  ...  x n
x   xi
x
n i 1
n
Find the mean number of home runs hit by
Barry Bonds for his first 16 seasons.
Find the mean number of home runs hit by
Barry Bonds for his first 15 seasons.
Important Concept: Mean is influenced by
outliers. Mean is not a RESISTANT MEASURE
Measuring center: the median
Def) Median M – midpoint of distribution; number such that
half the observations are larger and half are smaller
How to find a median:
1. Arrange values smallest to largest
2. Odd number of values, then M is the center value
3. Even number of values, then M is the mean of center two
values
Find the M for Bonds for his first 15 and
first 16 seasons
Is Median a resistant measure?
Median is a RESISTANT MEASURE
Properties of Mean and Median
1. Symmetric distribution results in mean and median values
that are very close together
2. Skew right – mean moves higher; median unchanged
3. Skew left- mean moves lower; median unchanged
Mean or Median
Total value of homes in a neighborhood.
Price of a typical home in a neighborhood.
Assignment
p. 40 #1.31-1.35 (20 minutes)
Measuring spread: the quartiles
Def) Range – difference between largest and smallest values
Def) First Quartile = Q1 – median of values to left of the
median M
Def) Third Quartile = Q3 – median of values to right to the
median M
Properties of Quartiles
1. Quartiles mark middle half
2. Q1 larger than 25% of values; Q3 larger than 75% of
values
Def) Interquartile Range (IQR) – range of quartiles
IQR = Q3-Q1
Find M, quartiles, IQR for Bonds’ and Aaron’s homerun data
Bond
16 19 24 25 33 33 34 34 37 37 40 42 46 49 73
Aaron
13 20 24 26 27 29 30 32 34 34 38 39 39 40 40 44 44 44 44 45 47
Determination of an Outlier
Def) Outlier – any value that falls more than 1.5IQR above Q3
or below Q1
Is Bond’s homerun year of 73 an outlier?
Five Number Summary – Displayed in a (Modified) Boxplot
Minimum, Q1, M, Q3, Maximum
Recall the data for Bond and Aaron and create a boxplot and a
modified boxplot
Modified Boxplot
1. Plot five number summary with outliers separate
2. Central box spans quartiles
3. Line in box marks median
4. Outliers are given points
5. Lines extend to min/max unless they are outliers
Technology Toolbox: Modified Boxplot
Compare Bonds, Aaron, and Ruth
Ruth = 54 59 35 41 46 25 47 60 54 46 49 46 41 34 22
Assignment
p. 48 #1.36-1.39 (30 minutes)
Measuring spread: the standard deviation
Def) Standard deviation – measurement of how far
observations are from the mean
Def) Variance = s2 = average of the squares of the deviations
of the observations from the mean
s
2
x


1
x
  x
2
Def) Standard deviation = s
2
x

2
n 1

 ...  x n  x

2
Metabolic rate is the rate at which the body consumes energy.
Given are the metabolic rates of seven men. The units are
calories per day.
1792
1897
1666
1439
1362
1614
1. Find the mean.
2. Find the deviations and the squared deviations
x
i
x

x
i
x

2
3. Find the variance = s2
4. Find the standard deviation (indicate units)
1460
Properties of the standard deviation
1. Used only when mean is chosen as a measure of center
When should mean be used as a measure of center?
2. S = 0 when there is no spread
3. S is not resistant to outliers
4. S is important later for symmetric (normal) distributions
Important.
ALWAYS PLOT YOUR DATA. A graph gives the best overall
picture of a distribution. Numerical measures of center and
spread give facts, but do not describe the whole shape.
Choosing mean and s versus 5 number summary
5 number summary for skewed data
Mean and s for symmetric distributions
Assignment
p. 52 #1.40-1.43 (20 minutes)
Stop here
Changing the unit of measurement
Linear Transformation – changes original variable x into a
new variable xnew by an equation
xnew = mx + b
Changes
1. b moves all values x up or down by b amount
2. m changes the size of the unit of measurement
What do these changes do to the 5 number summary, mean,
or the standard of deviation?
1. Create a stem plot of the player’s salaries and use your
calculator to find 5 number summary, s, and x
2. Suppose each player gets a bonus of $100,000. Write an
equation to find the new salaries. Find the values again.
Describe any changes.
3. Suppose there is a 10% raise given to each player. Write
and equation to find the new salaries. Find the values
again. Describe any changes.
Summary of Effects of a Linear Transformation
1. Multiplying by m multiplies mean, median, IQR, and
variance by m
2. Mean, median, and quartiles are changed by b
Assignment
p. 56 #1.44-1.46 (10 minutes)
Exercise 1.47 – Back to Back
Stemplot
What type of hotdog is
“better” for you?
Use side by side boxplots and
numerical summaries to
answer the question.
Assignment
p. 59 #1.48-1.50 (30 minutes)
Assignment Section 1.2
p. 62 #1.51-1.58 (80 minutes)
Assignment Chapter 1 Review
p. 66 #1.59-1.73 (80 minutes)