Transcript AP-Stats-2.5-Measure-of

AP Stats
BW
9/22
Which organization had the greatest total player earnings? Explain your reasoning.
Number of Players Separated into Earnings Ranges
Organization
$0-$500,000
$500,001$2,000,000
$2,000,001$6,000,000
$6,000,001$10,000,000
$10,000,001
+
MLB
399
207
189
73
56
MLS
316
5
0
0
0
NBA
31
166
147
58
42
NFL
760
758
274
70
15
NHL
85
448
177
17
0
NASCAR
27
13
33
5
0
PGA
115
117
29
2
0
Section2.5 – Measures of Position
SWBAT:
Identify and analyze patterns of distributions using shape,
center and spread.
Source: www.wilsonmar.com
Measures of Position
Standard Deviation is used to measure spread
associated with mean
When you know the mean AND standard deviation of
a data set, you can measure a specific data value’s
position in the data set with a standard score, or zscore.
Negative z-score: the x-value is below the mean
Positive z-score: the x-value is above the mean
Z = 0: the x-value is equal to the mean
Standard Score….. aka….. Z-Score
The z-score represents the number of standard deviations a
given value x falls from the mean μ.
Very unusual scores
Unusual scores
Usual scores
Z-Score - example
1) The mean speed of vehicles along a stretch of highway is
56 miles per hour with a standard deviation of 4 miles per
hour. You measure the speed of 3 cars as 62 mph, 47 mph
and 56 mph. Find the z-score corresponding to each
speed. What can you conclude?
1st car: x = 62; z= 1.5
2nd car: x = 47; z = -2.25
3rd car: x = 56; z = 0
If the distribution of the speeds is approximately normal,
the car traveling 47 miles per hour is traveling unusually
slowly because its speed has a z-score of -2.25.
Z-Score - example
2) The montly utility bills in a city have a mean of $70 and a
standard deviation of $8. Find the z-scores for bills of $60,
$71, and $92. What can you conclude?
x = 60; : z= -1.25
x = 71; z = 0.125
x = 92; z = 2.75
If the distribution of the bills is bell-shaped, then the $92
bill is unusually high as it’s standard z-score is 2.75
standard deviations above the mean.
Z-Score – Table 4: Standard Normal Distribution
The z-table will tell you what percentile the z-score falls into.
Using our z-scores from the phone bills, look up percentile
z= -1.25  0.1056  10.6th percentile
z = 0.125  between 0.5478 & 0.5517  55th percentile
z = 2.75  0.9970  98th percentile
Measures of Position: 5-number summary
The 5-number summary is used to measure position
associated with the median
Fractiles are numbers that partition, or divide, an
ordered data set into equal parts:
Fractiles
Summary
Symbols
Quartiles
Divide a data set into 4 equal parts Q1, Q2, Q3…Q4
Deciles
Divide a data set into 10 equal
parts
D1, D2, D3……D9
Percentiles
Divide a data set into 100 equal
parts
P1, P2, P3……P99
Interpreting Percentiles
Interpret the following ogives:
What test score represents the 72nd percentile? Interpret.
The 72nd percentile corresponds to a test score of 1700. This means that 72%
of students had an SAT score of 1700 or less.
Interpreting Percentiles, cont’d
At what percentile is a team that scores 40 touchdowns?
40 corresponds to the 50th percentile which means that 50% of the teams
scored 40 or fewer touchdowns.
QUARTILES
Divide an ordered data set into four approximately equal parts
Q1 (1st Quartile or lower quartile): ≈ ¼ of the data fall on or below the 1st quartile
Q2 (2nd Quartile or median): ≈ ½ of the data fall on or below the 2nd quartile
Q3 (3RD Quartile or upper quartile): ≈ ¾ of the data fall on or below the 3rd quartile
IQR (Interquartile range): the difference between the 3rd and 1st quartiles
Used to analyze the variation of the middle 50% of the data.
Also used to identify outliers.
IQR = Q3 – Q1
Q3 + 1.5(IQR)  outlier
Q3 + 3(IQR)  extreme outlier
Q1 – 1.5(IQR)  outlier
Q1 – 3(IQR)  extreme outlier
QUARTILES - Example
Finding Quartiles MANUALLY
The test scores of 15 employees enrolled in a CPR training
course are listed.
13, 9, 18, 15, 14, 21, 7, 10, 11, 20, 5, 18, 37, 16, 17
STEPS:
1. ORDER DATA
2. Find Q2 (Median)
3. Find Q1(Median of the lower half of the data)
4. Find Q3 (Median of the upper half of the data)
5. Calculate IQR
6. Calculate possible outlier values
QUARTILES - Example
The test scores of 15 employees enrolled in a CPR training
course are listed.
13, 9, 18, 15, 14, 21, 7, 10, 11, 20, 5, 18, 37, 16, 17
lower half
upper half
5 7 9 10 11 13 14 15 16 17 18 18 20 21 37
Q1
Q2
Q1 = 10
Q2 (median) = 15
Q3 = 18
IQR (18 – 10) = 8
Outliers - none: (18 + 1.5(8)) = 45 or (10 – 1.5(8)) = -17
Extreme outliers – none
Q3
QUARTILES, cont’d
Quartiles lead to FIVE-NUMBER SUMMARY
1.
2.
3.
4.
5.
Minimum entry
First Quartile
Median
Third Quartile
Maximum entry
The graphical representation of the 5-# summary is a
box-and-whisker plot
BOX-AND-WHISKERS PLOT, example
The test scores of 15 employees enrolled in a CPR training
course
5 7 9 10 11 13 14 15 16 17 18 18 20 21 37
Min: 5
Q1 = 10
Q2 (median) = 15
Q3 = 18
Max: 37
About half the scores lie between 10 and 18. Given the length of the right
whisker, 37 might be an outlier.
BOX-AND-WHISKERS PLOT, interpreting
Lowest wage is $22 and highest is $30 per day. The median wage is $25.
50% of the wages are between $23.80 and $27. The data is skewed right.
Lowest score was 54%, highest was 98%. The median score was 79%, with
the middle 50% of scores between 66% and 90%. The data is skewed left.
BOX-AND-WHISKERS PLOT, interpreting
Data set: 10, 12, 14, 18,
22, 23, 24
BOX-AND-WHISKERS PLOT – use Calculator!
1) Tuition costs (in thousands of dollars) for 25
liberal arts colleges are listed.
23, 25, 30, 23, 20, 22, 21, 15, 25, 24, 30, 25, 30,
20, 23, 29, 20, 19, 22, 23, 29, 23, 28, 22, 28
2) Tuition costs (in thousands of dollars) for 25
universities.
20, 26, 28, 25, 31, 14, 23, 15, 12, 26, 29, 24, 31,
19, 31, 17, 15, 17, 20, 31, 32, 16, 21, 22, 28
BOX-AND-WHISKERS PLOT, interpreting
HOMEWORK:
P 109. 3-10, 13, 15, 25, 27, 33, 35