Lecture 3 · Week 3 · Lind, Marchal & Wathen ch. 3 · about 55 min

Numerical Measures

Measures of location (mean, median, mode, weighted and geometric mean) and measures of dispersion (range, variance, standard deviation), plus Chebyshev's theorem, the Empirical Rule, and the mean and standard deviation of grouped data.

By the end you can

Compute and interpret the mean, median, and mode, and read skewness from their relative positions
Compute a weighted mean and the geometric mean, including average growth rates over time
Compute the range, variance, and standard deviation for a population and for a sample
Apply Chebyshev's theorem and the Empirical Rule to describe a distribution
Estimate the mean and standard deviation from grouped data

Chapter 2 described data with tables and graphs. This chapter describes quantitative data with numbers. Two families of numbers matter: measures of location (averages that mark the center) and measures of dispersion (numbers that describe the spread).

Two firms can both pay a mean salary of 80,000 dollars while one pays everyone between 70,000 and 90,000 and the other ranges from 40,000 to 120,000. The location is identical; the spread is not. Reporting only the average hides that difference, so a complete description always needs both a center and a spread.

The arithmetic mean

The arithmetic mean is the most widely reported average: add the values and divide by how many there are. The notation depends on whether the data are the whole population or a sample.

Population meanFormula 3-1Formula sheet →

\mu = \frac{\sum x}{N}

$\mu$: population mean
$x$: each observation
$N$: number of observations in the population

Use it when you have data for the entire population and want its arithmetic average.

Sample meanFormula 3-2Formula sheet →

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

$\bar{x}$: sample mean
$x$: each observation
$n$: number of observations in the sample

Use it when you have a sample and want its arithmetic average.

The mean needs at least interval-level data: differences between values must be meaningful.

There are 42 exits on I-75 in Kentucky, and the distances between consecutive exits add to 192 miles. Because every exit in the state is included, this is a population: $\mu = 192/42 = 4.57$ miles between exits.

Properties of the arithmetic mean

It requires at least interval-level data.
Every value in the data set enters the calculation.
It is unique: one data set has exactly one mean.
The deviations from the mean always cancel out: $\sum (x - \bar{x}) = 0$ .

The last property makes the mean the balance point of the data. For the values 3, 8, and 4 the mean is 5, and

$(3 - 5) + (8 - 5) + (4 - 5) = -2 + 3 - 1 = 0$ .

Median, mode, and skewness

The median is the midpoint of the values after they are ordered from smallest to largest. For an odd number of observations it is the middle value; for an even number it is the arithmetic mean of the two middle values (so it may not be one of the data values). It needs at least ordinal data and is not affected by extreme values.

Palm Aire condominium prices are 60,000; 65,000; 70,000; 80,000; and 275,000 dollars. The mean is 110,000, dragged up by the penthouse, but the median is 70,000, which is the typical price. If the penthouse had cost 1 million, the median would still be 70,000.

The mode is the value that occurs most frequently. It is the only measure of location that works for nominal data (for example, the most preferred bath oil in a survey). A data set can have no mode (all values different), or several: the ages 22, 26, 27, 27, 31, 35, 35 are bimodal with modes 27 and 35.

The shape of a distribution determines where the three measures sit:

Symmetric and mound-shaped: mean = median = mode, all at the center.
Positively skewed (right tail): mean is largest, then the median, then the mode. High outliers pull the mean up.
Negatively skewed (left tail): mean is smallest, then the median, then the mode. Low outliers pull the mean down.

The book’s examples: weekly income (positive skew) has mode 25, median 29, mean 60; tensile strength (negative skew) has mean 45, median 76, mode 80. If a distribution is strongly skewed, the mean alone is misleading; report the median (or both).

The weighted mean

When several observations share the same value, or when some observations matter more than others, the weighted mean is faster and more appropriate than adding repeated values one by one.

Weighted meanFormula 3-3Formula sheet →

\bar{x}_w = \frac{\sum (wx)}{\sum w}

$\bar{x}_w$: weighted mean
$x$: each value
$w$: the weight of that value

Use it when values repeat or are not equally important.

Wendy’s sold 3 medium drinks at 1.84, 4 large at 2.07, and 3 Biggie-sized at 2.40 dollars. Adding ten numbers gives a mean of 21.00/10 = 2.10 dollars. With weights, the same answer comes from

$\bar{x}_w = \dfrac{3(1.84) + 4(2.07) + 3(2.40)}{3 + 4 + 3} = \dfrac{21.00}{10} = 2.10$ dollars.

Grades are weighted by credit hours in exactly the same way.

The geometric mean and growth rates

The geometric mean is the measure for percentages, ratios, indexes, and growth rates that compound (each period builds on the previous one). The geometric mean of $n$ positive values is the $n$ th root of their product.

Geometric meanFormula 3-4Formula sheet →

GM = \sqrt[n]{x_1 \cdot x_2 \cdots x_n}

$GM$: geometric mean
$x$: each positive value
$n$: number of values

Use it when you average growth factors, ratios, or percentages instead of raw amounts.

Rate of increase over timeFormula 3-5Formula sheet →

GM = \sqrt[n]{\frac{\text{value at end}}{\text{value at start}}} - 1

$GM$: average compound growth rate
$n$: number of periods
$end and start$: values at the two dates

Use it when you want the average annual growth rate between two points in time.

For a growth rate over a stretch of time, divide the end value by the start value, take the $n$ th root, and subtract 1. Las Vegas grew from 258,295 people in 1990 to 613,599 in 2014 (24 years). The ratio is 2.3756, the 24th root is 1.0367, so the population grew about 3.67% per year.

Range and variance

A measure of location alone does not tell you how spread out the data are. Two assembly plants both produce a mean of 50 monitors per hour, but one varies from 48 to 52 and the other from 40 to 60. The mean is more trustworthy for the first plant.

The simplest measure of dispersion is the range.

RangeFormula 3-6Formula sheet →

Range = Maximum - Minimum

$Maximum$: largest value
$Minimum$: smallest value

Use it when you want a quick, easily understood measure of the total spread.

The range is easy but uses only two values; it says nothing about the values in between. The variance uses every value. It is the arithmetic mean of the squared deviations from the mean.

	Baton Rouge	Tucson
Mean	50.00	50.00
Median	50.00	50.00
Range	4.00	20.00
St. dev. s	1.22	6.95

Nine illustrative hours per plant, matching Chart 3-6: both plants average 50 monitors an hour, but Baton Rouge stays within 48 to 52 (range 4) while Tucson swings from 40 to 60 (range 20). Same mean, very different reliability.

Population varianceFormula 3-7Formula sheet →

\sigma^2 = \frac{\sum (x - \mu)^2}{N}

$\sigma^2$: population variance
$x$: each observation
$\mu$: population mean
$N$: population size

Use it when the data are the whole population.

Population standard deviationFormula 3-8Formula sheet →

\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}

$\sigma$: population standard deviation
$x$: each observation
$\mu$: population mean
$N$: population size

Use it when you want dispersion in the original units.

The steps are mechanical: find the mean, subtract it from each observation, square the differences, sum them, divide by $N$ . Beaufort County issued 348 citations over 12 months, so $\mu = 348/12 = 29$ . The squared deviations total 1,488, so $\sigma^2 = 1488/12 = 124$ and $\sigma = \sqrt{124} \approx 11.14$ citations.

Sample variance and standard deviation

When the data are a sample instead of a population, the mean becomes $\bar{x}$ and the denominator changes from $N$ to $n - 1$ .

Sample varianceFormula 3-9Formula sheet →

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

$s^2$: sample variance
$x$: each observation
$\bar{x}$: sample mean
$n$: sample size

Use it when the data are a sample.

Sample standard deviationFormula 3-10Formula sheet →

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

$s$: sample standard deviation
$x$: each observation
$\bar{x}$: sample mean
$n$: sample size

Use it when you want the sample spread in the original units.

Chebyshev’s theorem and the Empirical Rule

The standard deviation is a ruler for describing how much of a distribution lies near the mean. Two rules translate standard deviations into percentages.

Chebyshev’s theorem works for any distribution, whatever its shape: the proportion of values within $k$ standard deviations of the mean is at least $1 - 1/k^2$ , for any $k$ greater than 1.

k	Minimum proportion within k standard deviations
2	$1 - 1/4 = 0.75$ , at least 75%
3	$1 - 1/9 = 0.889$ , at least 88.9%
4	$1 - 1/16 = 0.9375$ , at least 93.8%
5	$1 - 1/25 = 0.96$ , at least 96%

Dupree Paint employees contribute a mean of 51.54 dollars to a profit-sharing plan with a standard deviation of 7.51 dollars. For $k = 3.5$ : at least $1 - 1/12.25 = 0.9184$ , about 92% of the contributions lie between $51.54 - 3.5(7.51) = 25.26$ and $51.54 + 3.5(7.51) = 77.83$ dollars.

The Empirical Rule is sharper, but only for a symmetrical, bell-shaped distribution: about 68% of observations lie within one standard deviation of the mean, about 95% within two, and practically all (99.7%) within three.

University Park Apartments: rental rates are bell-shaped, mean 500 dollars, standard deviation 20 dollars. Then about 68% pay between 480 and 520, about 95% between 460 and 540, and almost all between 440 and 560. Reversed, if a bell-shaped distribution runs from 70 to 130 (a range of 60), the standard deviation is roughly range divided by 6, or 10.

Grouped data: mean and standard deviation

When only a frequency distribution is available, each class is represented by its midpoint, $M$ . The result is an estimate of the mean, because individual values inside a class are unknown.

Arithmetic mean of grouped dataFormula 3-11Formula sheet →

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

$\bar{x}$: grouped sample mean
$f$: class frequency
$M$: class midpoint
$n$: total number of observations

Use it when you only have a frequency distribution.

Standard deviation of grouped dataFormula 3-12Formula sheet →

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

$s$: grouped sample standard deviation
$f$: class frequency
$M$: class midpoint
$\bar{x}$: grouped mean
$n$: total number of observations

Use it when you estimate dispersion from a frequency distribution.

For Applewood Auto Group, 180 vehicle profits in 8 classes give $\sum fM = 333,200$ , so the grouped mean is $333,200/180 = 1,851.11$ dollars, close to the raw-data mean of 1,843.17. The grouped standard deviation is 652.33 versus 643.63 from raw data: the estimate loses a little information, but stays very close.

Exam practice

Summary and review

Measures of location pinpoint the center: mean, median, mode, weighted mean, geometric mean.
Population mean $\mu = \sum x / N$ and sample mean $\bar{x} = \sum x / n$ . A population characteristic is a parameter; a sample characteristic is a statistic.
The mean uses every value, is unique, and is the balance point: $\sum (x - \bar{x}) = 0$ . It needs interval data and is sensitive to outliers.
The median is the ordered midpoint (average the two middle values when $n$ is even). It needs ordinal data and resists outliers, so it is preferred for skewed data.
The mode is the most frequent value; it works for nominal data and may be absent or bimodal.
Skewness landmarks: symmetric means mean = median = mode; positive skew means mode $\lt$ median $\lt$ mean; negative skew reverses the order.
Weighted mean $\bar{x}_w = \sum (wx) / \sum w$ ; the denominator is the sum of the weights.
Geometric mean $GM = \sqrt[n]{x_1 \cdots x_n}$ averages compounding factors; growth over time uses $\sqrt[n]{\text{end}/\text{start}} - 1$ . It is never above the arithmetic mean and needs positive values.
Range = maximum − minimum; quick, but based on only two values.
Variance is the mean squared deviation: divide by $N$ for a population, by $n - 1$ for a sample. Standard deviation is its square root, back in the original units.
Chebyshev’s theorem (any shape): at least $1 - 1/k^2$ of values lie within $k$ standard deviations of the mean, $k \gt 1$ .
Empirical Rule (bell-shaped only): about 68% within 1 standard deviation, 95% within 2, 99.7% within 3. Rough check: standard deviation about range/6.
Grouped data: $\bar{x} = \sum fM / n$ and $s = \sqrt{\sum f(M - \bar{x})^2 / (n - 1)}$ using class midpoints; both are estimates.
Exam traps to remember: $n - 1$ (not $n$ ) for sample variance; median for skewed data; sum of weights in the weighted mean; sample variance has squared units; Chebyshev works for any shape but the Empirical Rule needs a bell shape.