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Chapter 2: Modeling Distributions of Data 2.1 – Describing Location in a Distribution
Chapter 2: Modeling Distributions of Data
2.1 – Describing Location in a
Distribution
Learning Objectives
After this section, you should be able to:
• MEASURE position using percentiles
• INTERPRET cumulative relative frequency graphs
• TRANSFORM data
• DEFINE and DESCRIBE density curves
Measuring Position: Percentiles• The pth percentile of a distribution is the value with p percent of the
observations less than it.
• One way to describe the location of a value in a distribution is to tell
what percent of observations are less than it.
“at 70th percentile” not “in …”
Example #1
If Jenny is in a class of 25 students and makes an 86 on her test. How did she perform relative to the rest of her class? What percentile is she ranked in?
6 7
7 2334
7 5777899
8 00123334
8 569
9 03
6 7
7 2334
7 5777899
8 00123334
8 569
9 03
4th highest 21 lower scores
21
25= 84%
6 7
7 2334
7 5777899
8 00123334
8 569
9 03
Example #1
If Jenny is in a class of 25 students and makes an 86 on her test. How did she perform relative to the rest of her class? What percentile is she ranked in?
21
25= 84%
Her score was greater than 21 of the 25
observations.
Jenny is at the 84th percentile of the test
score distribution in her class.
Example #2
The stemplot shows the number of wins for each of the 30 Major
League Baseball teams in 2009.
5 9
6 2455
7 00455589
8 0345667778
9 123557
10 3
Rockies: 24
30= 80%
Yankees:29
30= 97%
*Can NOT be at 100th percentile with this method
5 9
6 2455
7 00455589
8 0345667778
9 123557
10 3
5 9
6 2455
7 00455589
8 0345667778
9 123557
10 3
Since 80% of the teams won less games,
they are at the 80th percentile.
Since 97% of the teams won less games,
they are at the 97th percentile.
Example #2
The stemplot shows the number of wins for each of the 30 Major
League Baseball teams in 2009.
5 9
6 2455
7 00455589
8 0345667778
9 123557
10 3
Indians:3
30= 10%
* 2 teams
at the
same
percentile
Since 10% of the teams won less games,
they are at the 10th percentile.
Cumulative Relative Frequency Graph
A cumulative relative
frequency graph displays the
cumulative relative frequency
of each class of a frequency
distribution.
Or called “ogive”
Example #3
The table shows the distribution of median household incomes for
the 50 states and the District of Columbia in a recent year.
Median Income
($1000s)
Frequency Relative frequency Cumulative
frequency
Cumulative relative
frequency
35 to < 40 1 1/51 = 0.020
40 to < 45 10 10/51 = 0.196
45 to < 50 14 14/51 = 0.275
50 to < 55 12 12/51 = 0.235
55 to < 60 5 5/51 = 0.098
60 to < 65 6 6/51 = 0.118
65 to < 70 3 3/51 = 0.059
Median Income
($1000s)
Frequency Relative frequency Cumulative
frequency
Cumulative relative
frequency
35 to < 40 1 1/51 = 0.020 1 1/51 = .020
40 to < 45 10 10/51 = 0.196 11 11/51 = 0.216
45 to < 50 14 14/51 = 0.275 25 25/51 = 0.490
50 to < 55 12 12/51 = 0.235 37 37/51 = 0.725
55 to < 60 5 5/51 = 0.098 42 42/51 = 0.824
60 to < 65 6 6/51 = 0.118 48 48/51 = 0.941
65 to < 70 3 3/51 = 0.059 51 51/51 = 1.000
Example #3
Cumulative
frequency
Cumulative relative
frequency
1 1/51 = .020
11 11/51 = 0.216
25 25/51 = 0.490
37 37/51 = 0.725
42 42/51 = 0.824
48 48/51 = 0.941
51 51/51 = 1.000
Example #3
a) California, with a median household
income of $57,445, is at what percentile?
Interpret this value.
About 79th percentile
In California, ~79% of household have an
income less than $57,445
b) What is the 25th percentile for this
distribution? What is another name for
this value?
25th percentile ~ $45,000
* Also known as 𝑸𝟏 *
Check your Understanding
The graph is a cumulative relative frequency graph showing the lifetimes
(in hours) of 200 lamps.
Measuring Position: z - scores
The z – score tells us how many standard deviations from the mean an observation falls,
and in what direction
Definition:
If x is an observation from a distribution that has known mean
and standard deviation, the standardized value of x is:
A standardized value is often called a z-score.
z x mean
standard deviationOr 𝑧 =
𝑥𝑖− ҧ𝑥
𝑠𝑥
** Unit = st. dev.
** Positive z score = above mean
** Negative z score = below mean
Example #4
The single-season home record for major league baseball has been set just three times since Babe Ruth hit
60 home runs in 1927. Roger Maris hit 61 in 1961, Mark McGwire hit 70 in 1998 and Barry Bonds hit 73 in
2001. In absolute sense, Barry Bonds had the best performance of these four players, because he hit the
most home runs in a single season. However, in a relative sense this may not be true. Baseball historians
suggest that hitting home run has been easier in some eras than others. This is due to many factors,
including quality of batters, quality of pitchers, hardness of the baseball, dimensions of ballparks, and possible
use of performance-enhancing drugs. To male fair comparison, we should see how these performances rate
relative to others hitters during the same year. Calculate the standardized score for each player and
compare.Year Player HR Mean SD
1927 Babe Ruth 60 7.2 9.7
1961 Roger Maris 61 18.8 13.4
1998 Mark McGwire 70 20.7 12.7
2001 Barry Bonds 73 21.4 13.2
Example #4
Year Player HR Mean SD
1927 Babe Ruth 60 7.2 9.7
1961 Roger Maris 61 18.8 13.4
1998 Mark McGwire 70 20.7 12.7
2001 Barry Bonds 73 21.4 13.2
Ruth: 𝑧 =60−7.2
9.7= 5.44
Maris: 𝑧 =61−18.8
13.4= 3.15
McGwire: 𝑧 =70−20.7
12.7= 3.88
Bonds: 𝑧 =73−21.4
13.2= 3.91
Example #4
Ruth: 𝑧 =60−7.2
9.7= 5.44
Maris: 𝑧 =61−18.8
13.4= 3.15
McGwire: 𝑧 =70−20.7
12.7= 3.88
Bonds: 𝑧 =73−21.4
13.2= 3.91
All player are above mean for their respective
year. However, Babe Ruth is the home run
champ, relatively speaking.
Class Example
• What variable could we measure about this class?
• # of sports played, # of siblings, # of phone contacts, # of texts per day, etc.
Put data in calculator and find 1 –Var Stats
• Find ҧ𝑥 𝑎𝑛𝑑 𝑠𝑥
Find your own z – score
Who is above the mean? Below mean?
Anyone with 𝑧 = 0
Example #5
In 2001, Arizona Diamondback Mark Grace’s home run total had a
standardized score of 𝑧 = −0.48. Interpret this value and calculate the
number of home runs he hit.
Grace hit 0.48 standard deviations below the mean of 21.4
home runs in 2001.
−0.48 =𝑥 − 21.4
13.2∙ 13.213.2 ∙
−6.336 = 𝑥 − 21.4
15 ≈ 𝑥
About 15 home runs
