DESCRIPTIVE STATISTICSSECTIONS 6 -9

Chapter 2

2.6 Percentiles

Quartiles are specific examples of percentiles. The first quartile is the same as the 25th percentile and the third quartile is the same as the 75th percentile.

The nth percentile represents the value that is greater than or equal to n% of the data.

EXAMPLE

Consider each of the following statements about percentiles.

Jennifer just received the results of her SAT exams. Her SAT Composite of 1710 is at the 73rd percentile. What does this mean?

Suppose you received the highest score on an exam. Your friend scored the second-highest score, yet you both were in the 99th percentile. How can this be?

EXAMPLE

Number of

Tickets

Frequency RF CRF

0 6 0.08 0.081 18 0.24 0.322 12 0.16 0.483 11 0.15 0.634 9 0.12 0.755 6 0.08 0.836 5 0.07 0.907 4 0.05 0.958 2 0.03 0.989 1 0.01 0.9910 1 0.01 1

The following data set shows the number of parking tickets received.

EXAMPLE

The following data set shows the number of parking tickets received.

Find and interpret the 90th percentile.

Find and interpret the 20th percentile.

Find the first quartile, the median, and the third quartile.

Construct a box plot.

2.6 IQR and outliers

Quartiles can also be used to help us spot potential outliers.

Inner Quartile Range:

A data point is a potential outlier is it is smaller than or greater than

EXAMPLE

Number of

Tickets

Frequency RF CRF

0 6 0.08 0.081 18 0.24 0.322 12 0.16 0.483 11 0.15 0.634 9 0.12 0.755 6 0.08 0.836 5 0.07 0.907 4 0.05 0.958 2 0.03 0.989 1 0.01 0.9910 1 0.01 1

The following data set shows the number of parking tickets received.

EXAMPLE

The following data set shows the number of parking tickets received.

Find the inner quartile range of the data set.

Do any of the data values appear to be outliers

2.7 Measures of Center

There are three common measures of the center of a data set:

Mean (average) represented by for a sample and by μ for the population. If x represents the data values and n and N represent the sample size and population size then:

(LoLN)

Median represented by M is the 50th percentile of the data set.

Mode is the most frequent data value. (can be more than one)

EXAMPLE

Find the mean median and mode of the following data set.

Use technology to find statistical information.

1. 4.5, 10, 1, 1, 9, 14, 4, 8.5, 6, 1, 9

EXAMPLE

Number of

Tickets

Frequency RF CRF

0 6 0.08 0.081 18 0.24 0.322 12 0.16 0.483 11 0.15 0.634 9 0.12 0.755 6 0.08 0.836 5 0.07 0.907 4 0.05 0.958 2 0.03 0.989 1 0.01 0.9910 1 0.01 1

The following data set shows the number of parking tickets received.

Find the mean, median, and mode.

Use technology to find statistical information.

2.9 Measures of Spread

The final statistics we would like to be able to find are measures that tell us how spread out the data is about the mean.

The two statistics that are most commonly used to measure spread are standard deviation and variation.

Standard deviation gives us another way to identify possible outliers: a data value might be an outlier if it is more than two standard deviations from the mean.

2.9 Calculating Standard Deviation and Variance

1. Calculate the deviation for each data value: of this tells you how far each data value is from the mean

2. Square each deviation: or

3. Add up the squares of the deviations: or

4. Find the variance:

i. Population:

ii. Sample:

5. Take the square root of the variance to find the standard deviation:

i. Population:

ii. Sample variance:

EXAMPLE

Find the standard deviation and variance of the data set assuming that it is a sample.

Use standard deviation to determine if any values are possible outliers.

Use technology to find statistical values.

1. 4.5, 10, 1, 1, 9, 17, 4, 8.5, 5, 1, 9

EXAMPLE

Number of

Tickets

Frequency RF CRF

0 6 0.08 0.081 18 0.24 0.322 12 0.16 0.483 11 0.15 0.634 9 0.12 0.755 6 0.08 0.836 5 0.07 0.907 4 0.05 0.958 2 0.03 0.989 1 0.01 0.9910 1 0.01 1

The following data set shows the number of parking tickets received.

Find the standard deviation and variance of the data set assuming that it is a sample.

Use standard deviation to determine if any values are possible outliers.

ExampleIn 2000 the mean age of a sample of females in the U.S. population was 37.8 years with a standard deviation of 21.8 years and the mean age of a sample of males was 35.3 with a standard deviation of 18.4 years.

In relation to the rest of their sex, which is older, a 48 year old woman or a 45 year old man?

Characterizing a distribution

1. Center, mean/median/mode

2. Skew

3. Spread

Characterizing a Data Distribution

Characterizing a Data Distribution

Characterizing a Data Distribution

Characterizing a Data Distribution

Characterizing a Data Distribution

Characterizing a Data Distribution

Characterizing a Data Distribution

Characterizing a Data Distribution

Characterizing a Data Distribution

Characterizing a Data Distribution

Example: For each distribution described below, discuss the number of peaks, symmetry, and amount of variation you would expect to find.

- The salaries of actors/actresses.

- The number of vacations taken each year.

- The weights of calculators stored in the math library – half are graphing calculators and half are scientific calculators.

HOMEWORK

2.13 #s 4a, b, c, 7, 10, 12, 13a, b, d, e, f, also construct a line graph for the data from Publisher A and Publisher B, 16a part i and iii, 16b, 21, 29, 30, 31