Lecture 1 · Week 1 · Lind, Marchal & Wathen ch. 1 · about 30 min

What Is Statistics?

Descriptive and inferential statistics, populations and samples, parameters and statistics, the core measures of center and spread, types of variables, the four levels of measurement, and the ethics of using data.

By the end you can

Define statistics and explain why knowledge of it matters.
Tell descriptive statistics apart from inferential statistics.
Distinguish population from sample and parameter from statistic.
Compute and interpret the mean, median, range, variance, standard deviation, and coefficient of variation.
Classify variables as qualitative or quantitative and discrete or continuous.
Identify the level of measurement: nominal, ordinal, interval, or ratio.
Recognize ethical issues and the role of business analytics.

What statistics is and why it matters

A statistic is one number that communicates a piece of information: the inflation rate is 2%, your grade point average is 3.5, the list price of a new car is 28,659 dollars. By itself a statistic is a very limited fact. It becomes powerful when it is part of a larger discussion about data.

Three reasons knowledge of statistics is important:

Data are collected everywhere and require statistical knowledge to be made useful.
Statistical techniques are used to make professional and personal decisions.
Whatever your career, you need statistics to understand the world and to be conversant in your field.

The scale of data is enormous: Google is estimated to process 20,000 terabytes of information per day; big retailers scan every purchase to manage distribution and marketing; police departments map crimes by location. All of these organizations turn raw data into intelligence for decisions, and statistics is the tool that does it.

Descriptive and inferential statistics

When you use statistics to produce information from data, there are two possibilities. They depend on the question asked and on the data you have.

Descriptive statistics take a mass of unorganized data and turn it into something readable. Examples from the text:

There are 46,837 miles of interstate highways in the United States; the system is 1% of all roads but carries more than 20% of the traffic. The longest is I-90 at 3,099 miles; the shortest is I-878 at 0.70 mile. Texas has the most interstate miles (3,232) and New York the most routes (28).
The average person spent 133.91 dollars on traditional Valentine’s Day merchandise in 2014, up 2.94 dollars from 2013. Men spent more than twice what women spent: 108.38 dollars versus 48.41 dollars.

Inference is how you decide with limited data. TV networks sample viewer preferences (Nielsen reported that 9.0% of a sample of households watched one program in a given week). The IRS tested a sample of tax preparation volunteers with three standard returns and found a 49% accuracy rate — errors on about half of the returns. In each case a sample result is used to reach a conclusion about a much larger population.

Brandon and Associates asked a sample of 1,960 consumers to try a new Boston Market chicken dinner; 1,176 said they would buy it. The firm uses this result to estimate acceptance among all potential customers. This is an example of:

Population, sample, and the notation you must know

A parameter is a numerical measure that describes a population. A statistic is a numerical measure that describes a sample and is used to estimate the parameter.

	Population (parameter)	Sample (statistic)
Mean	$\mu$ (Greek mu)	$\bar{x}$ (x-bar)
Standard deviation	$\sigma$ (Greek sigma)	$s$
Size	$N$	$n$
Proportion	$P$	$\hat{p}$

Population meanParameterFormula sheet →

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

$\mu$: population mean
$x$: each observation in the population
$N$: population size

Use it when you can observe every member of the population and want its average.

Sample meanStatisticFormula sheet →

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

$\bar{x}$: sample mean
$x$: each observation in the sample
$n$: sample size

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

Sample proportionStatisticFormula sheet →

\hat{p} = \frac{x}{n}

$\hat{p}$: sample proportion
$x$: number of observations with the attribute
$n$: sample size

Use it when you need the fraction of a sample that has some attribute (such as consumers who would buy a product).

The example behind the exercise from the chapter opener is the sale of Fitbit Flex 2 units at the Best Buy store in Collegeville, Pennsylvania. If your question is about that one store in that one month, the units sold are a population. If your question is about all Best Buy stores, or about future months, then that same month at that store is just one piece of evidence — a sample.

Center and spread: every formula you need

Chapter 1 is the big picture, but the midterm expects you to compute the basic summary measures with correct notation. Remember: Greek letters ( $\mu$ , $\sigma$ ) describe populations and Roman letters ( $\bar{x}$ , $s$ ) describe samples.

Mean

The mean is the arithmetic average — the balance point of the data. Add every value, divide by how many there are.

Population meanKey formulaFormula sheet →

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

$\mu$: population mean
$x$: each value in the population
$N$: population size

Use it when every member of the population is observed.

Sample meanKey formulaFormula sheet →

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

$\bar{x}$: sample mean
$x$: each value in the sample
$n$: sample size

Use it when you summarize a sample. The mean uses every observation, so a single extreme value can move it a lot.

Median

Order the data from smallest to largest. The median is the middle value.

If $n$ is odd, it is the value in position $(n + 1)/2$ .
If $n$ is even, it is the average of the two middle values.

Median positionFormula sheet →

\text{Median position} = \frac{n + 1}{2}

$n$: number of observations
$the data must be ordered firs$: the data must be ordered first

Use it when you want to locate the middle of ordered data. For even n, average the two middle values.

Range

RangeFormula sheet →

\text{Range} = \text{Maximum} - \text{Minimum}

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

Use it when you need a quick, crude measure of total spread. It uses only two observations, so it ignores everything in between.

Variance and standard deviation

The variance is the average squared distance of the observations from the mean. Squaring keeps positive and negative deviations from cancelling (they always sum to zero before squaring).

Population varianceKey formulaFormula 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 you have all population values and want their variability.

Sample varianceKey formulaFormula 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 you have a sample and want its variability.

Population standard deviationFormula sheet →

\sigma = \sqrt{\sigma^{2}}

$\sigma$: population standard deviation
$\sigma^{2}$: population variance

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

Sample standard deviationKey formulaFormula sheet →

s = \sqrt{s^{2}}

$s$: sample standard deviation
$s^{2}$: sample variance

Use it when you want spread in the original units, comparable with the mean.

Coefficient of variationFormula sheet →

CV = \frac{s}{\bar{x}} \times 100\%

$CV$: coefficient of variation
$s$: sample standard deviation
$\bar{x}$: sample mean

Use it when you compare variability of variables measured in different units or with very different means; the result is a percentage.

Skewed data: use the median

Consider six annual salaries (in thousands of dollars) at a small firm: 20, 22, 24, 26, 28, 130. The mean is $250/6 \approx 41.67$ thousand, but the median is $(24 + 26)/2 = 25$ thousand. The single large salary drags the mean up into a region where no employee actually sits. This distribution is right-skewed: the long tail of large values pulls the mean above the median.

Types of variables

There are two basic types of variables.

Examples: gender, beverage preference, type of vehicle owned, state of birth, eye color. Qualitative data are usually summarized in bar charts and pie charts.

Examples: the balance in your checking account, gigabytes of data used on your phone plan, the life of a car battery (42 months), the number of people employed by a company. Quantitative variables come in two kinds:

Discrete: can assume only certain values, and there are “gaps” between values. Discrete variables are counted. A house can have 3 or 4 bedrooms but not 3.56; a store sells 326 or 421 units in an hour, not 326.4 units; a section has 25 students, another 42, another 18.
Continuous: can assume any value within a range. Continuous variables are measured. Air pressure in a tire, the weight of a shipment of tomatoes, ounces of raisins in a box of cereal, the duration of a flight. GPA is continuous: a student’s GPA could be 3.2576952, usually rounded to 3.258.

Levels of measurement

The level of measurement tells you how data may be summarized, presented, and analyzed. There are four levels, from lowest (nominal) to highest (ratio). Getting this wrong makes an analysis meaningless: assigning M&M colors the codes 1 to 6 and reporting an “average color” of 3.56 tells you nothing, and the same is true of the “mean finish” of 4.5 in a race. Codes for categories are labels, not numbers.

Examples: M&M colors (blue 24%, green 20%, orange 16%, yellow 14%, red 13%, brown 13% of a bag), gender, state of birth (states can be coded alphabetically 1 to 50 so software can count them). No meaningful arithmetic: if Alabama is 1 and Alaska is 2, it does not follow that Alabama + Alaska = Arizona (3).

Examples: state rankings for “best business climate” (Florida 1st, Utah 2nd, Texas 3rd, Georgia 4th, Indiana 5th); team standings; instructor ratings on the scale inferior, poor, good, excellent, superior. In a finance class of 60 students, ratings were superior 6, good 26, average 16, poor 9, inferior 3; about 43.3% (26/60) rated the instructor good. We cannot say whether the gap between superior and good equals the gap between poor and inferior.

Examples: the Fahrenheit temperature scale. Boston winter highs of 28, 31, and 20 degrees can be ranked, and a 5-degree interval is the same everywhere (from 10 to 15 as from 50 to 55). But 0°F does not mean “no heat,” and 20°F is not “twice as warm” as 10°F. Dress sizes are another interval scale: going from size 10 to 12 adds exactly 2 inches to bust, waist, and hips, but size 0 is not “zero material,” and ratios of sizes are meaningless. If the distances make sense but the ratios do not, the scale is interval.

Examples: wages, weight, height, units produced, distance between offices, income. If you have zero dollars you have no money, and 50 dollars per hour is exactly twice 25 dollars per hour. In the textbook’s father-son income table, the senior Lahey (80,000 dollars) earns twice his son (40,000 dollars), and in the Rho family the son (120,000 dollars) earns twice the father (60,000 dollars).

Level	Order?	Meaningful intervals?	True zero and ratios?	Examples
Nominal	No	No	No	Jersey numbers, gender, M&M color
Ordinal	Yes	No	No	Class rank, team standings, instructor rating
Interval	Yes	Yes	No	Temperature (°F/°C), dress size, SAT scores
Ratio	Yes	Yes	Yes	Salary, weight, height, units sold

Ethics, integrity, and business analytics

High-profile failures — Bernie Madoff’s Ponzi scheme, and the financial misrepresentations at Enron and Tyco — were built on misreported business and financial information. Investors lost all or nearly all of their money when the true numbers emerged. The lesson for anyone who works with data:

Practice statistics with integrity and honesty, and “do the right thing” when collecting, organizing, summarizing, analyzing, and interpreting numerical information.
Keep “an independent and principled point-of-view” when analyzing and reporting findings.
Report information that truly reflects performance, and report product defects or risks that may harm people.

As an informed consumer of statistics you should question a report based on data that do not fairly represent the population, a report that omits relevant statistics, one that uses an incorrect choice of statistical measures, or a presentation designed to introduce bias and mislead.

Business analytics is the use of data and statistical analysis to support a story or narrative about a company’s business — “what makes us profitable?”, “how will our customers respond to a change in marketing?” It combines statistical knowledge with software (in this course, mainly Excel) to summarize, organize, analyze, and present findings. The textbook’s running example is Applewood Auto Group: 180 vehicle sales described by buyer age, repeat-customer status, dealership location, vehicle type, and profit. Software summarizes profit as mean 1,843.17 dollars, median 1,882.50 dollars, and range from 294 to 3,292 dollars.

An analyst finds that a sampled survey about a new product heavily oversampled one age group, so the sample does not represent the population. The company would still like a favorable headline. What is the ethical course of action?

Exam practice

Summary and review

Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting data to assist in making more effective decisions. A single number such as a GPA is a statistic.
Descriptive statistics organize, summarize, and present data. Inferential statistics estimate a property of a population from a sample.
A population is the entire set of objects of interest; a sample is a part of it. A parameter describes a population ( $\mu$ , $\sigma$ , $N$ ); a statistic describes a sample ( $\bar{x}$ , $s$ , $n$ ). Whether a set is a population or a sample depends on the question, not on its size.
Mean $= \sum x / n$ is the balance point and uses all data. Median is the middle of ordered data; it is resistant to extremes and is the better “typical value” for skewed data (incomes, prices, profits).
Sample variance $s^{2} = \sum (x - \bar{x})^{2} / (n - 1)$ — divide by $n - 1$ , never by $n$ . Standard deviation $s = \sqrt{s^{2}}$ is spread in the original units. CV $= (s / \bar{x}) \times 100\%$ compares variability across different units.
Qualitative variables are nonnumeric (gender, brand); quantitative variables are numeric. Quantitative variables are discrete if counted with gaps (bedrooms, cars per hour) and continuous if measured (weight, GPA, flight time).
Levels of measurement: nominal (labels, no order), ordinal (order, unknown gaps), interval (equal intervals, no true zero, e.g. temperature in °F), ratio (true zero and meaningful ratios, e.g. salary, weight). Codes are not measurements.
Ethics: collect, analyze, and report data with integrity, honesty, and an independent point of view; as a consumer, question unrepresentative samples, missing statistics, wrong measures, and biased presentations. Business analytics uses statistics plus software to support a business narrative (for example, the Applewood Auto Group profit summary: mean 1,843.17, median 1,882.50, range 294 to 3,292 dollars).