Measure of Position

• Looking at individual values: Where does it lie among the sample (or population)– standard score (or z score)

– quartiles, deciles, and percentiles

• 25%-75% SAT scores– Princeton 1370 – 1560

– Villanova 1180 – 1340

Standard Score (z score)

• How many standard deviations from the mean is a sample?

• Z scores less that –2.00 or greater that 2.00 are outliers.

s

Xxz n

For example

• 25 55 59 59 63 71 71 74 80 80 80 83 84 84 87 88 95 95 100 100

• Mean : 77• Standard deviation = 18

mean

mean +

mean + 2

mean + 3mean - 3

mean - 2

mean - 1

Quartiles

• Separates the sorted list into four parts separated by three boundaries

min lower quartile

median maxupper quartile

25% 25% 25% 25%

• If the median separates the list into two halves– The lower quartile is the median of the lower half

– The upper quartile is the median of the upper half

For example

• 25 55 59 59 63 71 71 74 80 80 80 83 84 84 87 88 95 95 100 100

Deciles

• 25 55 59 59 63 71 71 74 80 80 80 83 84 84 87 88 95 95 100 100

min median max

10% 10% 10%

Percentile

• Percentile value of a value x• Where does x sit in a set of data

• For example:

• 25 55 59 59 63 71 71 74 80 80 80 83 84 84 87 88 95 95 100 100

• What is the percentile of:

number of values less than 100

total number of values

x

Finding the value at the kth Percentile

• n values in our sample• Data has to be sorted

• Compute L, the position of the kth percentile

• If L is a whole number, the percentile is midway between L and L+1

• If L is a decimal, round up; the percentile is the Lth value

nk

Lth

100

For example

• 25 55 59 59 63 71 71 74 80 80 80 83 84 84 87 88 95 95 100 100

• What is the value at the

Relationships

• The 1st quartile is the same as the 25th percentile• The 3rd quartile is the same as the 75th percentile• The first decile is the same as the 10th percentile

Live example

• What are the mean and standard deviation

• What are the minimum, lower quartile, median, upper quartile, and maximum

• What is the 4th decile• What is the 35% percentile?• What is South’s z value• What is South’s percentile?

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Your turn

• What are the mean and standard deviation

• What are the minimum, lower quartile, median, upper quartile, and maximum

• What is the 4th decile?• What is the 35% percentile?• What is Rob’s z score?• What is Rob’s percentile?

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