MATH 1401 · Dr. Barry Monk

Measures of Spread (Continued) and Measures of Position

Sections 3.2 & 3.3

We pick up Section 3.2 where the last lecture left off.

Definition

Standard Deviation

Since variance is calculated using squared deviations, its units are the squared units of the data. However, it’s often more practical to use a measure of spread with the same units as the data. This is achieved by taking the square root of the variance, resulting in the standard deviation.

Sample Standard Deviation

s = \sqrt{s^{2}}

Population Standard Deviation

σ = \sqrt{σ^{2}}

Example

Recall that the variance for the San Francisco monthly temperatures was $σ^{2}$ = 14.083 squared degrees.

	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
San Francisco	51	54	55	56	58	60	60	61	63	62	58	52

The standard deviation is

σ = \sqrt{14.083} = 3.753

The Empirical Rule

When a data set has a bell-shaped histogram, it is often possible to use the standard deviation to provide an approximate description of the data using a rule known as The Empirical Rule.

Approximately 68% of the data will be within one standard deviation of the mean.

μ - σ

μ + σ

Approximately 95% of the data will be within two standard deviations of the mean.

μ - 2 σ

μ + 2 σ

All, or almost all, of the data will be within three standard deviations of the mean.

μ - 3 σ

μ + 3 σ

μ

σ

Example

Example: The Empirical Rule

Projections for the population percentage aged 65 and over for each state is given. Describe the population using the Empirical Rule.

14.1	14.3	14.4	17.8	12.0	14.9	12.6	13.7	12.8	13.8	13.7	12.4	13.8
14.1	13.3	14.3	16.0	8.1	11.5	14.1	10.2	12.4	13.4	15.6	12.8	13.9
12.3	14.1	15.3	13.0	13.6	10.5	12.4	13.5	13.9	10.7	11.5	14.3	12.7
13.1	12.2	12.4	15.0	12.6	13.6	13.7	15.5	14.6	9.0	12.2	14.0

Solution:

Note that the histogram is bell-shaped, so the Empirical Rule applies. We find the mean and standard deviation.

Mean:

μ = 13.249

Standard Deviation:

σ = 1.683

Online calculator

Example

Example: The Empirical Rule

We compute six quantities:

μ - σ = 13.25 - 1.683 = 11.57

μ + σ = 13.25 + 1.683 = 14.93

Approximately 68% of the data values are between 11.57 and 14.93.

μ - 2 σ = 13.25 - 2 (1.683) = 9.88

μ + 2 σ = 13.25 + 2 (1.683) = 16.62

Approximately 95% of the data values are between 9.88 and 16.62.

μ - 3 σ = 13.25 - 3 (1.683) = 8.20

μ + 3 σ = 13.25 + 3 (1.683) = 18.30

Almost all of the data values are between 8.20 and 18.30.

Recap

Section 3.2 — Measures of Spread

Standard Deviation

The square root of the variance, so it carries the same units as the data.

s = \sqrt{s^{2}}

σ = \sqrt{σ^{2}}

The Empirical Rule

For a bell-shaped data set, approximately 68% of the data lies within one standard deviation of the mean, approximately 95% within two, and all, or almost all, within three.

Section 3.3 · Objective 1 of 4

Compute and interpret $z$ -scores

Definition

$z$ -Score

Let $x$ be a value from a population with mean $μ$ and standard deviation $σ$ . The $z$ -score of $x$ is:

z = \frac{x - μ}{σ}

Interpretation:

The $z$ -score of an individual data value tells how many standard deviations that value is from its population mean.

For example, a value one standard deviation above the mean has a $z$ -score of $z = 1$ and a value two standard deviations below the mean has a $z$ -score of $z = - 2$ .

A man and a woman standing outdoors in tall grass under a pale overcast sky. The man is nearer the camera on the left and the woman a little behind him on the right; he is noticeably the taller of the two.

Example:

A study shows the average height for U.S. men is 69.4 inches ( $σ = 3.1$ ) and for women is 63.8 inches ( $σ = 2.8$ ). Who is taller relative to their gender: a 73-inch man or a 68-inch woman?

Z_{Man’s Height} = \frac{x - μ}{σ} = \frac{73 - 69.4}{3.1} = 1.16

Z_{Woman’s Height} = \frac{x - μ}{σ} = \frac{68 - 63.8}{2.8} = 1.50

The woman is taller, relative to the population of women’s heights.

Section 3.3 · Graphical interpretation

The Two Heights on One Scale

Adult men

μ = 69.4

Adult women

μ = 63.8

One shared scale

z

Finding a Value with a Given $z$ -Score

To find a value from a given $z$ -score, multiply the $z$ -score by the standard deviation, then add the result to the mean.

x = μ + z \cdot σ

Example:

Data from the National Health and Examination Survey shows that for U.S. adults, the mean cholesterol level is $μ = 200$ mg/dL ( $σ = 44$ ). What is the cholesterol level for someone with a $z$ -score of 1.5?

Solution:

x = μ + z \cdot σ = 200 + 1.5 (44) = 266

The Empirical Rule & $z$ -Scores

A long airport concourse in low evening light, windows down one side and gate signs overhead. Two travellers with wheeled luggage walk away from the camera in the middle distance.

Check Your Understanding

The Wall Street Journal reports that the average maximum distance from check-in to the gate at top U.S. airports is $μ = 0.79$ miles ( $σ = 0.42$ miles).

The maximum distance at the Seattle-Tacoma International Airport is 0.57 miles. What is the $z$ -score for this distance?

z = \frac{0.57 - 0.79}{0.42} = - 0.52

What distance would have a $z$ -score of 1?

x = 0.79 + 1 (0.42) = 1.21

miles

Quartiles

We’ve learned how to compute the mean and median of a data set as measures of the center. Sometimes, it is useful to compute measures of position other than the center. Quartiles divide a data set into four approximately equal pieces.

Q_{1}

The first quartile, $Q_{1}$ , separates the lowest 25% of the data from the highest 75%.

Q_{2}

The second quartile, $Q_{2}$ , divides the data so that 50% is below and 50% is above, essentially representing the median.

Q_{3}

The third quartile, $Q_{3}$ , separates the lowest 75% of the data from the highest 25%.

Percentiles

Percentiles divide a data set into hundredths.

p

and

(100 - p)

For any number $p$ between 1 and 99, the $p^{th}$ percentile separates the lowest $p$ 50% of the data from the highest $(100 - p)$ 50%.

p

Definition

Interquartile Range

One method for detecting outliers involves a measure called the Interquartile Range (IQR). The interquartile range is found by subtracting the first quartile from the third quartile: $IQR = Q_{3} - Q_{1}$ .

Any data value beyond the outlier boundaries may be an outlier.

Lower Outlier Boundary

Q_{1} - 1.5 (IQR)

Upper Outlier Boundary

Q_{3} + 1.5 (IQR)

An empty classroom. Rows of black tablet-arm chairs stand unoccupied in front of tall daylit windows, with no students present.

Example

Example: Outliers

Online calculator

The table shows the number of students absent from school each day in January. Identify any outliers in the data.

65	67	71	57	51	49	44	41	59	49	42
56	45	77	44	42	45	46	100	59	53	51

Solution:

IQR = Q_{3} - Q_{1} = 59 - 45 = 14

Lower Boundary:

Q_{1} - 1.5 (IQR) = 45 - 1.5 (14) = 24

Upper Boundary:

Q_{3} + 1.5 (IQR) = 59 + 1.5 (14) = 80

The value 100 is an outlier because it is greater than 80.

Recap

Section 3.3 — Measures of Position

z

-Scores

The $z$ -score of an individual data value tells how many standard deviations that value is from its population mean.

z = \frac{x - μ}{σ}

Quartiles & Percentiles

Quartiles divide a data set into four approximately equal pieces; percentiles divide it into a hundred.

Q_{1}

Q_{2}

Q_{3}

Five-Number Summary

The five-number summary of a data set consists of the median, the first quartile, the third quartile, the smallest value, and the largest value.

Outliers

One method for detecting outliers involves a measure called the Interquartile Range (IQR).

IQR = Q_{3} - Q_{1}

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0.00	0.03	0.08	0.16	0.17	0.20	0.29	0.56	0.70	0.79	0.83	0.92	1.22	1.30	1.48
1.64	1.72	1.90	2.37	2.84	3.06	3.12	3.21	3.29	3.54	3.57	3.58	3.71	4.13	4.17
4.27	4.37	4.64	4.89	4.94	5.54	5.59	6.10	6.61	7.96	8.87	8.91	11.02	12.75	13.68

Measures of Spread (Continued) and Measures of Position

Compute the standard deviation of a population and a sample

Standard Deviation

The Standard Deviation is Not Resistant

Use The Empirical Rule to summarize data that are unimodal and approximately symmetric

The Empirical Rule

Example: The Empirical Rule

Example: The Empirical Rule

Section 3.2 — Measures of Spread

Measures of Position

Compute and interpret $z$ -scores

$z$ -Score

Example:

The Two Heights on One Scale

Finding a Value with a Given $z$ -Score

The Empirical Rule & $z$ -Scores

Check Your Understanding

Compute the quartiles of a data set

Quartiles

Suppose…

Example: Computing Quartiles

Visualizing the Quartiles

Percentiles

Visualizing Percentiles

Check Your Understanding

Compute the five-number summary for a data set

Five-Number Summary

Five-Number Summary with Technology

Understand the effects of outliers

Outliers

Interquartile Range

Example: Outliers

Section 3.3 — Measures of Position

You Are Ready For:

Measures of Spread (Continued) and Measures of Position

Compute the standard deviation of a population and a sample

Standard Deviation

The Standard Deviation is Not Resistant

Use The Empirical Rule to summarize data that are unimodal and approximately symmetric

The Empirical Rule

Example: The Empirical Rule

Example: The Empirical Rule

Section 3.2 — Measures of Spread

Measures of Position

Compute and interpret z-scores

z-Score

Example:

The Two Heights on One Scale

Finding a Value with a Given z-Score

The Empirical Rule & z-Scores

Check Your Understanding

Compute the quartiles of a data set

Quartiles

Suppose…

Example: Computing Quartiles

Visualizing the Quartiles

Percentiles

Visualizing Percentiles

Check Your Understanding

Compute the five-number summary for a data set

Five-Number Summary

Five-Number Summary with Technology

Understand the effects of outliers

Outliers

Interquartile Range

Example: Outliers

Section 3.3 — Measures of Position

You Are Ready For:

Compute and interpret $z$ -scores

$z$ -Score

Finding a Value with a Given $z$ -Score

The Empirical Rule & $z$ -Scores