MATH 1401 · Dr. Barry Monk

Measures of Center

Section 3.1

Definition

Notation: Population Versus Sample

A population includes all individuals of interest, while a sample is a smaller group taken from the population.

The mean is calculated the same way for both, but the notation is different.

Notation:

Population Mean:

μ

Sample Mean:

\bar{x}

Sum of Values:

\sum x

Definition

Computing the Mean

Given a list of numbers $x_{1}, x_{2}, x_{3}, \dots$ , the sample mean is computed as:

\bar{x} = \frac{\sum x}{n}

… and the population mean is computed as:

μ = \frac{\sum x}{N}

Median of a Data Set

The median is a measure of center that divides an ordered data set in half, with half the values below it and half above it. The method to calculate the median varies depending on whether the number of observations is even or odd.

Work inward from both ends, discarding one value from each side at a time.

If $n$ is odd, the median is the middle number.

If $n$ is even, the median is the average of the two middle numbers.

A student alone at a classroom desk, writing on a paper with a pen, in an otherwise empty room.

Example

Example: Computing the Mean and Median

During a semester, a student took five exams. The population of exam scores is 78, 83, 92, 68, and 85. Find the mean and median of the exam scores.

Mean

\sum x = 78 + 83 + 92 + 68 + 85 = 406

N = 5

μ = \frac{\sum x}{N} = \frac{406}{5}

μ = 81.2

Median

Arrange the data values in increasing order.

68 78 83 85 92

The median is the middle number, 83.

A quiet hospital room with made-up beds, a drip stand and a chair by the window: a recovery ward, not an operating room.

Example

Example: Median with Even Number of Values

Eight patients undergo a new surgical procedure, and the number of days spent in recovery for each is as follows. Find the median number of days in recovery.

20 15 12 27 13 19 13 21

Solution:

Arrange the data values in increasing order.

12 13 13 15 19 20 21 27

The median is the average of the two middle numbers.

\frac{15 + 19}{2} = 17

Definition

Population Variance

σ^{2} = \frac{\sum {(x - μ)}^{2}}{N}

Example

Example: Population Variance

Compute the population variance for the San Francisco temperatures.

Step 1: Compute the population mean

μ

μ = \frac{\sum x}{N} = \frac{690}{12} = 57.5

Step 2: For each population value compute the difference between the value and the mean:

	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
$x$	51	54	55	56	58	60	60	61	63	62	58	52
$(x - μ)$	−6.5	−3.5	−2.5	−1.5	0.5	2.5	2.5	3.5	5.5	4.5	0.5	−5.5

The differences from the mean always add up to zero, which is why we square them.

Example

Example: Population Variance

Recall: $μ = 57.5$ , $N = 12$

	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
$x$	51	54	55	56	58	60	60	61	63	62	58	52
$(x - μ)$	−6.5	−3.5	−2.5	−1.5	0.5	2.5	2.5	3.5	5.5	4.5	0.5	−5.5
${(x - μ)}^{2}$	42.25	12.25	6.25	2.25	0.25	6.25	6.25	12.25	30.25	20.25	0.25	30.25

Step 3: Square the differences.

Step 4: Add the squared differences:

\begin{array}{l} \sum {(x - μ)}^{2} = 42.25 + 12.25 + 6.25 + 2.25 + 0.25 + 6.25 \\ + 6.25 + 12.25 + 30.25 + 20.25 + 0.25 + 30.25 = 169 \end{array}

Step 5: Divide the sum by the population size:

σ^{2} = \frac{\sum {(x - μ)}^{2}}{N} = \frac{169}{12} = 14.083 .

The variance is in squared units: here, degrees squared.

Definition

Sample Variance

When computing sample variance, the sample mean is used for deviations. Deviations using the sample mean are typically smaller than those using the population mean.

If we divided by $n$ when calculating sample variance, it would generally be smaller than the population variance. To correct this, we divide the sum of squared deviations by $n - 1$ instead.

Sample Variance

s^{2} = \frac{\sum {(x - \bar{x})}^{2}}{n - 1}

Example

Example: Sample Variance

A new type of battery is being tested for laptop computers. The lifetimes, in hours, of six batteries, are 3, 4, 6, 5, 4, 2. Find the sample variance of the lifetimes.

Solution:

Step 1: Compute the sample mean.

\bar{x} = \frac{\sum x}{n} = \frac{24}{6} = 4

Step 2: Square each difference from the mean, then add the squares.

$x$	3	4	6	5	4	2
$x - \bar{x}$	−1	0	2	1	0	−2
${(x - \bar{x})}^{2}$	1	0	4	1	0	4

\sum {(x - \bar{x})}^{2} = 1 + 0 + 4 + 1 + 0 + 4 = 10

Step 3:

s^{2} = \frac{\sum {(x - \bar{x})}^{2}}{n - 1} = \frac{10}{6 - 1} = \frac{10}{5} = 2

Divide by $n - 1 = 5$ , not by $n = 6$ .

Check Your Understanding

Population or Sample?

For each one, is this a population or a sample? Do we divide by $N$ or by $n - 1$ ?

A teacher computes the variance of the test scores for all 28 students in her class.

Population

Every student in the class is included, so divide by $N = 28$ .

A city planner records the commute times of 40 randomly chosen residents to estimate the variance of commute times for the whole city.

Sample

40 residents stand in for the whole city, so divide by $n - 1 = 39$ .

A running club records the finishing times of all 9 of its members in Saturday’s race.

Population

All 9 members are included, so divide by $N = 9$ .

Recap · Section 3.2

Two measures of spread

Range

The range of a data set is the difference between the largest value and the smallest value.

Population variance

Variance measures how far, on average, data values are from the mean.

σ^{2} = \frac{\sum {(x - μ)}^{2}}{N}

Sample variance

Measured from the sample mean, and divided by $n - 1$ instead of $N$ .

s^{2} = \frac{\sum {(x - \bar{x})}^{2}}{n - 1}

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Measures of Center

Compute the mean and median of a data set

Mean of a Data Set

Notation: Population Versus Sample

Computing the Mean

Median of a Data Set

Example: Computing the Mean and Median

Example: Median with Even Number of Values

The Complete Statistics Calculator

Running 1-Var Stats in the Calculator

1-Var Stats Command

Example: Mean and Median with 1-Var Stats

Compare properties of the mean and median

The Mean is More Influenced by Extreme Values

Example: The Median is Resistant

Mean, Median, and the Shape of a Data Set

Skewed to the right, skewed to the left, or approximately symmetric?

Find the mode of a data set

Mode of a Data Set

Example: Mode of a Data Set

Mode – Measure of Center?

Mode for Qualitative Data

Three measures of center

Measures of Spread

San Francisco vs. St. Louis

San Francisco vs. St. Louis

Compute the range of a data set

Range of a Data Set

Compute the variance of a population and a sample

The Variance

Population Variance

Example: Population Variance

Example: Population Variance

Sample Variance

Example: Sample Variance

Population or Sample?

Two measures of spread

You Are Ready For:

	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
St. Louis	30	35	44	57	66	75	79	78	70	59	45	35

	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
San Francisco	5151	54	55	56	58	60	60	61	6363	62	58	52
St. Louis	3030	35	44	57	66	75	7979	78	70	59	45	35