Statistics Class 6

February 8th 2012

Reveiw

The number of incidents in which police were needed for a sample of 10 schools in Allegheny County is 7, 37, 3, 8, 48, 11, 6, 0, 10, 3. Using the range rule of thumb do you consider 48 incidents at one school unusual?

Measures of Variation

Empirical Rule (68-95-99.7) for data with a Bell-Shaped Distribution.

• About 68% of all values fall within 1 standard deviations of the mean.

• About 95% of all values fall within 2 standard deviations of the mean.

• About 99.7% of all values fall within 3 standard deviations of the mean.

IQ scores have a bell-shaped distribution with a mean of 100 and a standard deviation of 15. What percentage of IQ scores are between 70 and 130?

Coefficient of Variation

• The Coefficient of Variation or CV for a set of nonnegative sample or population data, expressed as a percent, describes the standard deviation relative to the mean, and is given by the following:

Sample

Population

Compare the variation in the heights of men to the variation in weights of men, using these sample results obtained from Data Set 1 in Appendix B: for men, the heights yield a mean of 68.34 in. and standard deviation 3.02 in; the weights yield a mean of 172.55 lbs. and a standard deviation of 26.33 lbs.

heights: weights:

Measures of Relative Standing

A z score ( or standardized value) is the number of standard deviations that a given value x is above or below the mean. The z score is calculated by using one of the following:

Sample

Population

We now consider a comparison of two individual data values as we try to determine which is more extreme: the 76.2in. Height of a man or the 237.1 lb weight of a man. Compare those two data values by finding their corresponding z scores. Use these sample results:

Heights have mean in. and standard deviation in.

Weights have mean lb and standard deviation lb.

z scores, Unusual Values, and Outliers

Ordinary values: .

Unusual values: or .

The data values that are unusual, i.e. have a z score less than -2 or greater than 2, are called outliers.

A student scored 65 on a calculus test that had a mean of 50 and a standard deviation of 10: she scored a 30 on a history test with a mean of 25 and a standard deviation of 5. Compare her relative positions on the two tests.

Percentiles

Percentiles are measures of location, denoted , , …, which divide a set of data into 100 groups with about 1% of the values in each group.

For example, the 50th percentile denoted has about 50% of the data values below and above it.

PercentilesFinding the Percentile of a Data Value

(Round result to the nearest whole number)

PercentilesFinding the Percentile of a Data ValueConsider this list of 35 sorted movies budgets taken from a simple random sample.

Find the percentile for the value of $52 million.

4.5 5 6.5 7 20 20 29 30 35 4040 41 50 52 60 65 68 68 70 7070 72 74 75 80 100 113 116 120 125

132 150 160 200 225

PercentilesFinding the Percentile of a Data ValueFind the percentile for the value of $52 million.

4.5 5 6.5 7 20 20 29 30 35 4040 41 50 52 60 65 68 68 70 7070 72 74 75 80 100 113 116 120 125

132 150 160 200 225

4.5 5 6.5 7 20 20 29 30 35 4040 41 50 52 60 65 68 68 70 7070 72 74 75 80 100 113 116 120 125

132 150 160 200 225

4.5 5 6.5 7 20 20 29 30 35 4040 41 50 52 60 65 68 68 70 7070 72 74 75 80 100 113 116 120 125

132 150 160 200 225

4.5 5 6.5 7 20 20 29 30 35 4040 41 50 52 60 65 68 68 70 7070 72 74 75 80 100 113 116 120 125

132 150 160 200 225

So the budget amount of $52 million is at the 37th percentile.

PercentilesConverting a percentile to a Data ValueTo convert a percent to a data value we use the following formula

kth percentile.

PercentilesConverting a percentile to a Data ValueTo convert a percent to a data value we use the following formula

Tricky PartIf L is not a whole number then round it up, then is the Lth value counting from the lowest.If L is a whole number, then is midway between the Lth value and the next value in the sorted set of data. So .

PercentilesConverting a percentile to a Data Value Lets find the value of the Movie Budget that is at the 90th percentile.

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

. L is not a whole number so we round up to the 32nd value.

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

We round up to the 32nd value. We count to the 32nd position in our chart.

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

We round up to the 32nd value. We count to the 32nd position in our chart.

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

We round up to the 32nd value. We count to the 32nd position in our chart.So the value of $150 million is at the 90th percentile, that is about 90% of the movies have budgets below $150 million.

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

PercentilesConverting a percentile to a Data Value Now find the value of the Movie Budget that is at the 60th percentile.

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

. L is a whole number so is midway between the 21st and 22nd positions on our chart

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

. L is a whole number so is midway between the 21st and 22nd positions on our chart. So

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

So So the value of $71 million is at the 60th Percentile.

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

A teacher gives a 20-point test to 10 students. The scores are shown here. Find the percentile rank of a score of 12.18, 15, 12, 6, 8, 2, 3, 5, 20, 10

Using the same data as the previous slide find the value that corresponds to the 60th percentile.

QuartilesQuartiles are measures of location, denoted , which divide a set of data into four groups with about 25% of the values in each group.

• (First Quartile): Separates the bottom 25% of the sorted values from the top 75%.

• (Second Quartile): Same as the median; separates the bottom 50% of the sorted values form the top 50%

• (Third Quartile): Separates the bottom 75% of the sorted values from the top 25%.

• Or use your calculator!

QuartilesHow to Find a Quartile Find the value of the first Quartile of the movie budget data.Hint: Finding is the same as find

So the rounding up we get 9.

So the first Quartile of the movie budget data is $35 million

5-Number Summary and BoxplotFor a set of data, the 5-number summary consists of the minimum value, the first quartile , the median (or second quartile ), the third Quartile , and the maximum value.

A boxplot (or box-and whisker diagram) is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile , the median and the third quartile

5-Number Summary and BoxplotFinding a 5-Number Summary Use the Movie budget amounts to find a 5 digit summary.

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

5-Number Summary and BoxplotFinding a 5-Number Summary Use the Movie budget amounts to find a 5 digit summary.

min = 4.5max = 225 = $35 millionMedian = $60 million =$113 million

4.5 5 6.5 7 20 20 29 30 35 40

40 41 50 52 60 65 68 68 70 70

70 72 74 75 80 100 113 116 120 125

132 150 160 200 225

5-Number Summary and BoxplotConstructing a Boxplot1. Find the 5-Number Summary consisting of the minimum value, , the

median, , and the maximum value.

median, , and the maximum value.2. Construct a scale with values that include the minimum and maximum

data values.

data values.3. Construct a box (rectangle) extending from to , and draw a line in the

box at the median value.

box at the median value.4. Draw lines extending outward from the box to the minimum and

maximum data values.

box at the median value.4. Draw lines extending outward from the box to the minimum and

maximum data values.

5-Number Summary and BoxplotConstruct a Boxplot for the Movie Budget DataStep 1 is already complete since we have already found the 5-Number summarymin = 4.5max = 225 = $35 millionMedian = $60 million =$113 million

Now construct a box-plot for our 10 test scores

18, 15, 12, 6, 8, 2, 3, 5, 20, 10

Homework!

3-2: 7-15 odd, 29, 31, 413-3: 9-15 odd, 22-293-4: 1-17 odd

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