AP Stats BW 9/22

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

Which organization had the greatest total player earnings? Explain your reasoning.

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 z-score.

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.

Usual scores

Unusual scoresVery unusual 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= 1.52nd car: x = 47; z = -2.253rd 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= -1.25x = 71; z = 0.125x = 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 percentilez = 0.125 between 0.5478 & 0.5517 55th percentilez = 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

>95% OR <5% UNUSUALLY HIGH/LOW

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.

QUARTILESDivide 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 - ExampleFinding 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 DATA2. 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 IQR6. Calculate possible outlier values

QUARTILES - ExampleThe 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

5 7 9 10 11 13 14 15 16 17 18 18 20 21 37

Q2 Q3Q1

lower half upper half

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

QUARTILES, cont’dQuartiles lead to FIVE-NUMBER SUMMARY

1. Minimum entry2. First Quartile3. Median4. Third Quartile5. Maximum entry

The graphical representation of the 5-# summary is a box-and-whisker plot

BOX-AND-WHISKERS PLOT, exampleThe test scores of 15 employees enrolled in a CPR training course5 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