Chapter 4

Numerical Methods for Describing Data

Population characteristic -• Fixed value about a population• Typical unknown

Suppose we want to know the MEAN length of all the fish in Lake Lewisville . . .Is this a value that is known?Can we find it out?

At any given point in time,

how many values are

there for the mean length of fish in the

lake?

Statistic -• Value calculatedcalculated from a sampleSuppose we want to know the MEAN length of all the fish in Lake Lewisville.

What can we do to estimate this unknown population characteristic?

Measures of Central Tendency• Mode – the observation that occurs

the most often– Can be more than one mode

– If all values occur only once – there is no mode

– Not used as often as mean & median

Measures of Central TendencyMedian - the middle value of the data;

it divides the observations in half

To find: list the observations in numerical order

even is if values middle two the of average

odd is is value middle single median sample

n

n

Where n = sample size

Suppose we catch a sample of 5 fish from the lake. The lengths of the fish (in inches) are listed below. Find the median length of fish.

33 4 4 5 5 8 8 10 10

The numbers are in order & n is odd – so

find the middle observation.

The median length of fish is 5

inches.

Suppose we caught a sample of 6 fish from the lake. The median length is …

33 4 4 5 5 6 6 8 8 10 10

The numbers are in order & n is even – so find the middle two

observations.

The median length is 5.5

inches.

Now, average these two values.

5.5

Measures of Central TendencyMean is the arithmetic average.

– Use to represent a population mean

– Use x to represent a sample mean

n

xx

Formula:

is the capital Greek letter sigma – it means to sum the values that

follow

Population characteristic

statistic

is the lower case Greek letter mu

Suppose we caught a sample of 6 fish from the lake. Find the mean length of the fish.

33 4 4 5 5 6 6 8 8 10 10

To find the mean length of fish - add the observations and

divide by n.

61086543

6x

x (x - x)

3

4

5

6

8

10

Sum

What is the sum of the deviations from the mean?

Now find how each observation deviates from the mean.

0

Will this sum always equal zero?

YES

This is the deviation from the mean.

3-6-3

-2

-1

0

2

4

Find the rest of the deviations from the

mean

The mean is considered the balance point of the distribution because it “balances” the positive and negative deviations.

Imagine a ruler with pennies placed at 3”, 4”, 5”, 6”, 8” and 10”.

To balance the ruler on your finger, you would need to place your finger at the mean of 6.

The mean is the balance point of a distribution

What happens to the median & mean if the length of 10 inches was 15 inches?

33 4 4 5 5 6 6 8 8 15 15

The median is . . .

5.5

The mean is . . .

61586543

6.833

What happened?

What happens to the median & mean if the 15 inches was 20?

33 4 4 5 5 6 6 8 8 20 20

The median is . . .

5.5

The mean is . . .

62086542

7.667

What happened?

Some statistics that are not affected by extreme values . . .

Is the median resistant affected by extreme values?

Is the mean affected by extreme values?

NO

YES

Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width of 1.)

Mean =Median =

3 5 6 10 6 7 7 8 4 5 6 4 7 5 9 9 8 7 6 8

6.5

Calculate the mean and median.

6.5

Look at the placement of the mean and median in this symmetrical distribution.

Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width 1.)

Mean =Median =

5.56.8

Calculate the mean and median.

Look at the placement of the mean and median in this

skewed distribution.

3 5 6 10 15 7 3 3 4 5 6 4 12 5 3 4 8 13 11 9

Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width of 1.)

Mean =Median =8.5

7.75

Calculate the mean and median.

Look at the placement of the mean and median in this

skewed distribution.

3 5 6 10 10 7 10 8 9 5 6 4 9 10 9 9 10 7 10 8

Recap:• In a symmetrical distribution, the

mean and median are equal.• In a skewed distribution, the mean is

pulled in the direction of the skewness.

• In a symmetrical distribution, you should report the mean!

• In a skewed distribution, the median should be reported as the measure of center!

Trimmed mean:Purpose is to remove outliers from a

data set

To calculate a trimmed mean:• Multiply the percent to trim by n• Truncate that many observations

from BOTH ends of the distribution (when listed in order)

• Calculate the mean with the shortened data set

Mean = 23.8

Find the mean of the following set of data.

12 14 19 20 22 24 25 26 26 50

10%(10) = 1

So remove one observation from each side!

228

2626252422201914

Tx

Find a 10% trimmed.

60% of the sample was satisfied with their cell phone service.

6.0159ˆ p

What values are used to describe categorical data?Suppose that each person in a sample of 15 cell phone users is asked if he or she is satisfied with the cell phone service.

Here are the responses:Y N Y Y Y N N Y Y N Y Y Y N NWhat would be the possible responses?

Find the sample proportion of the people who answered “yes”:

n

successes ofnumber ˆ p

Pronounced p-hatThe population proportion is

denoted by the letter p.

Why is the study of variability Why is the study of variability important?important?• There is variability in virtually everything

• Allows us to distinguish between usual & unusual values

• Reporting only a measure of center doesn’t provide a complete picture of the distribution.

Does this can of soda contain exactly 12 ounces?

20 30 40 50 60 70

20 30 40 50 60 70

20 30 40 50 60 70

Notice that these three data sets all have the same mean and median (at 45), but they have very different amounts of variability.

Measures of VariabilityThe simplest numeric measure of variability is range.

Range = largest observation – smallest

observation20 30 40 50 60 70

20 30 40 50 60 70

20 30 40 50 60 70

The first two data sets have a range of 50 (70-20) but the third data set has a much smaller range of 10.

1

22

n

xxs

Measures of VariabilityAnother measure of the variability in a data set uses the deviations from the mean (x – x).

Remember the sample of 6 fish that we caught from the lake . . .They were the following lengths:

3”, 4”, 5”, 6”, 8”, 10”The mean length was 6 inches. Recall that we calculated the deviations from the mean. What was the sum of these deviations?

Can we find an average deviation?

What can we do to the deviations so that we

could find an average?

Degree of freedom

The estimated average of the deviations squared is called the variance.

Population variance is

denoted by 2.

When calculating sample variance, we use degrees of freedom (n – 1) in the denominator instead of n because this tends to produce better estimates.

Degrees of freedom will be revisited again in Chapter 8.

Suppose that everyone in the class caught a sample of 6 fish from the lake. Would each of our samples contain the same

fish?Would our mean lengths be

the same?

The samples would also have different ranges!

x (x - x) (x - x)2

3 -3

4 -2

5 -1

6 0

8 2

10 4

Sum 0

What is the sum of the deviations squared?

Remember the sample of 6 fish that we caught from the lake . . .Find the variance of the length of fish.

Divide this by 5.

First square the deviations

Finding the average of the deviations would

always equal 0!

9

4

1

0

4

1634 s2 = 6.5

Measures of VariabilityThe square root of variance is called standard deviation.

A typical deviation from the mean is the standard deviation.

s2 = 6.8 inches2 so s = 2.608 inches

The fish in our sample deviate from the mean of 6 by an average of 2.608 inches.

Calculation of standard deviation of a sample

1

2

n

xxs

Population standard deviation is denoted by (where n is used in the

denominator).

The most commonly used measures of center and

variability are the mean and standard deviation, respectively.

Measures of VariabilityInterquartile range (iqr) is the range of the middle half of the data.

Lower quartile (Q1) is the median of the lower half of the dataUpper quartile (Q3) is the median of the upper half of the data

iqr = Q3 – Q1

What advantage does the interquartile range have over the

standard deviation?

The iqr is resistant to extreme values

The Chronicle of Higher Education (2009-2010 issue) published the accompanying data on the percentage of the population with a bachelor’s or higher degree in 2007 for each of the 50 states and the District of Columbia.

21 27 26 19 30 35 35 26 47 26 27 30 24 29 22 24 29 20 20 27 35 38 25 31 19 24 27 27 23 34 25 32 26 24 22 28 26 30 23 25 22 25 29 33 34 30 17 25 23 34 26

Find the interquartile range for this set of data.

21 27 26 19 30 35 35 26 47 26 27 30 24 29 22 24 29 20 20 27 35 38 25 31 19 24 27 27 23 34 25 32 26 24 22 28 26 30 23 25 22 25 29 33 34 30 17 25 23 34 26

First put the data in order & find the median.

17 19 19 20 20 21 22 22 22 23 23 23 24 24 24 24 25 25 25 25 25 26 26 26 26 26 26 27 27 27 27 27 28 29 29 29 30 30 30 30 31 32 33 34 34 34 35 35 35 38 47

26

Find the lower quartile (Q1) by finding the median of the lower

half.

24

Find the upper quartile (Q3) by finding the median of the upper

half.

30

iqr = 30 – 24 = 6

Another graph- Boxplots

What are some advantages of boxplots?

• ease of construction• convenient handling of outliers• construction is not subjective (like

histograms)• Used with medium or large size data

sets (n > 10)• useful for comparative displays

BoxplotsWhen to Use Univariate numerical data

How to construct a Skeleton Boxplot– Calculate the five number summary– Draw a horizontal (or vertical) scale– Construct a rectangular box from the lower

quartile (Q1) to the upper quartile (Q3)– Draw lines from the lower quartile to the

smallest observation and from the upper quartile to the largest observation

To describe – comment on the center, spread, and shape of

the distribution and if there is any unusual features

Use for moderate to

large data sets. Don’t use with data sets of n <

10.

The five-number summary is the minimum value, first quartile,

median, third quartile, and maximum value

Remember the data on the percentage of the population with a bachelor’s or higher degree in 2007 for each of the 50 states and the District of Columbia.

17 19 19 20 20 21 22 22 22 23 23 23 24 24 24 24 25 25 25 25 25 26 26 26 26 26 26 27 27 27 27 27 28 29 29 29 30 30 30 30 31 32 33 34 34 34 35 35 35 38 47

10 20 30 40 50

Percentages

First draw a scaleDraw a box from Q1 to Q3

Draw a line for the median

Draw lines for the whiskers

Modified boxplotsTo display outliers:• Identify mild & extreme outliers

An observation is an outliers if it is more than 1.5(iqr) away from the nearest quartile.

An outlier is extreme if it is more than 3(iqr) away from the nearest quartile.

• whiskers extend to largest (or smallest) data observation that is not an outlier

iqrQiqrQ 5.1and5.1 31

iqrQiqrQ 3and3 31

Modified boxplots are generally preferred because they provide more

information about the data distribution.

Remember the data on the percentage of the population with a bachelor’s or higher degree in 2007 for each of the 50 states and the District of Columbia.

17 19 19 20 20 21 22 22 22 23 23 23 24 24 24 24 25 25 25 25 25 26 26 26 26 26 26 27 27 27 27 27 28 29 29 29 30 30 30 30 31 32 33 34 34 34 35 35 35 38 47

10 20 30 40 50

Percentages

First, draw the scale, box and the line for

the median

Draw lines for the whiskers

Next calculate the fences for outliers.24-1.5(6) =

15

30+1.5(6) = 39

30+3(6) = 48

There is one outlier at the upper end at the distribution, but none at the lower end. Is

it extreme?

Place a solid dot for the outlier

To describe:The distribution of percent of the population with a bachelor’s degree or higher for the U.S. states and District of Columbia is positively skewed with an outlier at 47%. The median percentage is at 26% with a range of 30%.

Symmetrical boxplots Approximately symmetrical boxplot

Skewed boxplot

Notice that all 3 boxplots are identical,

but their corresponding

histograms are very different. Can you

determine the number of modes from a

boxplot?

Notice that the range of the lower half and

the range of the upper half of this distribution are

approximately equal so we can say that it

is approximately symmetrical.However, the range of

the two halves of this distribution are

definitely different sizes, so it would be

skewed in the direction of the

longest side.

The 2009-2010 salaries of NBA players published on the web site hoopshype.com were used to construct the comparative boxplot of salary data for five teams.

Discuss the similarities

and differences.

Interpreting Center & VariabilityChebyshev’s Rule –

The percentage of observations that are within k standard deviations of the mean is at least

where k > 1

%1

11002

k

If k = 2, then at least 75% of the observations are within 2 standard deviations of the mean.

%75%2

11100

2

This rule can be used with any distribution – no matter it’s

shape!

For a sample of families with one preschool child, it was reported that the mean child care time per week was approximately 36 hours with a standard deviation of approximately 12 hours.

Using Chebyshev’s rule, at least 75% of the sample observations must be between 12 and 60 hours (within 2 standard deviations of the mean).

At most, what percent of the observations are greater than 72 hours?

At least 89% of the observations are between 0 & 72 hours. Since time can’t be negative, at most 11% of the observations are above

72 hours.

Input the following command into a graphing calculator in order to graph a normal curve with a mean of 20 and standard deviation of 3.

Y1 = normalpdf(X,20,3)(Window x: [10,30] y: [0,0.2])

Use the command 2nd trace, 7 to find the area under the curve for the: (Round to 3 decimal places.)

Lower limit: 17 Upper limit: 23 Area: ________Lower limit: 14 Upper limit: 26 Area: ________Lower limit: 11 Upper limit: 29 Area: ________

What’s my area?

Graph a normal curve with a mean of 50 and standard deviation of 5.

Y1 = normalpdf(X,50,5) (x: [30,70] y: [0,0.1])

Find the area under the curve for the following:

Lower limit: 45 Upper limit: 55 Area: ________Lower limit: 40 Upper limit: 60 Area: ________Lower limit: 35 Upper limit: 65 Area: ________

What’s my area?

What pattern do you notice?

Interpreting Center & VariabilityEmpirical Rule-

• Approximately 68% of the observations are within 1 standard deviation of the mean

• Approximately 95% of the observations are within 2 standard deviation of the mean

• Approximately 99.7% of the observations are within 3 standard deviation of the mean

Can ONLY be used with distributions that are mound

shaped!

68%95%99.7%

The height of male students at PWSH is approximately normally distributed with a mean of 71 inches and standard deviation of 2.5 inches.

a)What percent of the male students are shorter than 66 inches?

b) Taller than 73.5 inches?

c) Between 66 & 73.5 inches?

About 2.5%

About 16%

About 81.5%

Measures of Relative StandingZ-score

A z-score tells us how many standard deviations the value is from the mean.

deviation standard

mean-valuescore-z

One example of standardized score.

What do these z-scores mean?

-2.3

1.8

-4.3

2.3 standard deviations below the mean

1.8 standard deviations above the mean

4.3 standard deviations below the mean

Sally is taking two different math achievement tests with different means and standard deviations. The mean score on test A was 56 with a standard deviation of 3.5, while the mean score on test B was 65 with a standard deviation of 2.8. Sally scored a 62 on test A and a 69 on test B. On which test did Sally score the best?

714.15.3

5662

z

She did better on test A.

Z-score on test A Z-score on test B

429.18.2

6569

z

Measures of Relative StandingPercentiles

A percentile is a value in the data set where r percent of the observations fall AT or BELOW that value

In addition to weight and length, head circumference is another measure of health in newborn babies. The National Center for Health Statistics reports the following summary values for head circumference (in cm) at birth for boys.

Head circumference (cm)

32.2 33.2 34.5 35.8 37.0 38.2 38.6

Percentile 5 10 25 50 75 90 95

What percent of newborn boys had head circumferences greater than 37.0 cm?

10% of newborn babies have head circumferences bigger than what value?

25%

38.2 cm