Describing Location in a Distribution

Measuring Position: Percentiles

• Here are the scores of 25 students in Mr. Pryor’s statistics class on their first test:

79 81 80 77 73 83 74 93 78 80 75 67 73 77 83 86 90 79 85 83 89 84 82 77 72• Set this up into a stemplot.

Stemplot

Percentile

• Jenny had an 86, what percentile is she?• Percentile – the pth percentile of a distribution is

the value with p percent of the observations less than it.

• Hint: Norman, who earned a 72 has only one person who scored below his score. Since only 1 of the 25 students in the class is below his 72, his percentile is computed as follows: 1/25 = 0.04, or 4%. So Norman scored at the 4th percentile.

Find Percentiles

• Find the percentiles of: • Katie, who scored 93.• The two students who earned scores of 80.

The Rules…

• When we shift data by adding or subtracting the same constant to every data value, measures of location/center change, but measures of spread stay the same.

• When we rescale data by multiplying or dividing by the same constant to every data value, measures of location/center AND measures of spread change.

In Other Words …

• Adding (or Subtracting) a Constant:– Adding the same number to each observation• Adds a to measures of center and location (mean,

median, quartiles, percentiles), but• Does not change the shape of the distribution or

measures of spread (range, IQR, standard deviation)

• Multiplying (or Dividing) by a Constant:– Multiplying or dividing each observation by the

same number b • Multiplies (divides) measures of center and location

(mean, median, quartiles, percentiles) by b• Multiplies (divides) measures of spread (range, IQR,

standard deviation by b but• Does NOT change the shape of the distribution

Practice

• Suppose the average score on a national test is 100 with a standard deviation of 10. If each score is increased by 5, what happens to the mean and standard deviation?

• Suppose the average score on a national test is 100 with a standard deviation of 10. If each score is increased by 50%, what happens to the mean and standard deviation?

Back to Standard Deviation …

• The standard deviation tells us how the whole collection of values varies, so it’s a natural ruler for comparing an individual to a group.

• As the most common measure of variation, the standard deviation plays a crucial role in how we look at data.

Z-Scores.

• We compare individual data values to their mean, relative to their standard deviation using the following formula:

• We call the resulting values standardized values, denoted as z. They can also be called z-scores.

y yz

s

How do we compare??

The z-score helps us make comparisons. This z-score is simply a measure of “how many standard deviations from the mean”

zvalue mean

stdev

What can z-scores tell us??

A negative z-score tells us that the data value is below the mean.

A positive z-score tells us that the data value is above the mean

A z-score of 0 tells us a data value is right at the mean.

The further a z-score is from 0, the more unusual it is…

Example

• The average height of a man is 5’10”, with a standard deviation of 3”.

• The average height of a woman is 5’4” with a standard deviation of 4”.

• Who is more extreme in height, a woman who is 5’8” or a man who is 6’2”?

• CDC Anthropometric Reference Data for Children and Adults: U.S. Population, 1999–2002 - Page 20, Table 19

Another Example

Yet Another Example

Who is the best athlete?

Example

80 students each took an English test and a math test. Here are some statistics:

• Math– Average = 75– S.D. = 1.47

• English – Average = 80– S.D. = 1.19

Plots of the tests

Plots of the tests

Who did better?

• Suppose that both classes are graded on a curve

• The top English score was 85• The top math score was 80• Who did better?

The ruler

• To answer the question of “who scored higher, relative to the rest of the class”, we need standardize each variable to see how many “standard deviations” away each is from their means…

How to standardize?

• Subtract mean from the value, and then divide by the standard deviation.

• So, we shift the data’s mean, then scale it down

• This expresses the distance in comparable terms..

Back to our example

• For the top English score:(85-80)/1.19 = 4.20

• For the top math score:(80-75)/1.47 = 3.40

• So, the english student did better.

To formalize this…

• Expressing distance in standard deviations standardizes the performances.

• We subtract the mean score, then divide by the standard deviation

• Again, this is known as a “z-score”

Slide 6- 26

Benefits of Standardizing

• Standardized values have been converted from their original units to the standard statistical unit of standard deviations from the mean.

• Thus, we can compare values that are measured on different scales, with different units, or from different populations.