Stat201 exam 1

Statistics

The science of learning from data and using data to understand the world.

Data

Recorded values along with their context.

Context

The background information needed to understand what the data values represent.

The W's

Who, What, When, Where, Why, and How; questions used to understand the context of data.

Who

The individuals or cases about whom or which data are collected.

What

The variables recorded about each individual.

Why

The purpose for which the data were collected.

Population

The entire group of individuals or objects we want information about.

Sample

A subset of the population from which data are actually collected.

Why use a sample?

To learn about or make inferences about a larger population.

Representative sample

A sample that reflects the important characteristics of the population.

Randomness

Using chance in selecting observations helps reduce bias and obtain a representative sample.

Are random events necessarily equally likely?

No. Random means the outcome is uncertain, not that every possible outcome has the same probability.

Variable

A characteristic recorded for each individual or case.

Categorical variable

A variable whose values identify categories or groups.

Quantitative variable

A variable whose values are numerical measurements or counts for which arithmetic makes sense.

Identifier

A variable used to uniquely identify individuals; it may consist of numbers but is not quantitative.

Ordinal variable

A categorical variable whose categories have a meaningful order.

ZIP code

Categorical, even though it uses numbers, because the numbers are labels rather than measurements.

Student ID

Identifier, not quantitative.

Number of siblings

Quantitative because it is a numerical count.

Income

Quantitative because it is a numerical measurement with meaningful arithmetic.

Class year

Categorical/ordinal because Freshman, Sophomore, Junior, Senior are ordered categories.

Frequency

The number of observations in a category.

Relative frequency

The proportion or percentage of observations in a category.

Frequency table

A table displaying counts for categories of a categorical variable.

Relative frequency table

A table displaying proportions or percentages for categories.

Bar chart

A display for a categorical variable using separated bars to show counts or percentages.

Pie chart

A display for one categorical variable showing each category as a portion of the whole.

Pareto chart

A bar chart whose categories are ordered from greatest to smallest frequency.

Histogram

A display of the distribution of a quantitative variable using adjacent bins.

Bar chart vs histogram

Bar charts display categorical data and have separated bars; histograms display quantitative data grouped into intervals and normally have touching bars.

Dotplot

A display for quantitative data in which each observation is represented by a dot.

Stem-and-leaf display

A quantitative display that shows the shape of a distribution while preserving individual data values; especially useful for smaller datasets.

Distribution

The pattern of values of a variable, including its shape, center, spread, and unusual features.

SOCS

A useful reminder for describing quantitative distributions: Shape, Outliers/unusual features, Center, Spread.

Mode

A peak or local high point in a distribution.

Unimodal

A distribution with one main peak.

Bimodal

A distribution with two main peaks.

Multimodal

A distribution with more than two peaks.

Symmetric distribution

A distribution whose two sides are approximately mirror images around the center.

Right-skewed distribution

A distribution with a longer tail extending toward larger values.

Left-skewed distribution

A distribution with a longer tail extending toward smaller values.

Right skew: mean vs median

The mean is generally greater than the median because large values pull the mean to the right.

Left skew: mean vs median

The mean is generally less than the median because small values pull the mean to the left.

Mean

The arithmetic average: add all observations and divide by the number of observations.

Mean formula

x̄ = Σx / n.

Median

The middle observation after the data are ordered; if n is even, average the two middle observations.

Resistance

A statistic is resistant if extreme observations have relatively little effect on it.

Is the mean resistant?

No. Extreme observations can strongly affect the mean.

Is the median resistant?

Yes. Extreme observations usually have relatively little effect on the median.

Best center for symmetric data

Mean, especially when there are no major outliers.

Best center for skewed data

Median.

Best center when there are major outliers

Median.

Range

Maximum − Minimum.

Standard deviation

A measure of how spread out observations are around the mean.

Small standard deviation

Observations are relatively tightly clustered around the mean.

Large standard deviation

Observations are relatively spread out from the mean.

Is standard deviation resistant?

No. Extreme observations can substantially affect it.

Best spread for symmetric data

Standard deviation, especially when there are no major outliers.

Best spread for skewed data

IQR.

Best spread when there are outliers

IQR.

Q1

The first quartile; approximately the 25th percentile.

Median percentile

The median is the 50th percentile.

Q3

The third quartile; approximately the 75th percentile.

Percentile

A value below which a given percentage of observations fall.

82nd percentile interpretation

About 82% of observations are at or below that value.

What does Q1 tell you?

About 25% of observations are at or below Q1.

What does Q3 tell you?

About 75% of observations are at or below Q3 and about 25% are at or above Q3.

Five-number summary

Minimum, Q1, Median, Q3, Maximum.

IQR

Interquartile range; the spread of the middle 50% of observations.

IQR formula

IQR = Q3 − Q1.

What percentage of observations lies between Q1 and Q3?

Approximately 50%.

Lower outlier fence

Q1 − 1.5(IQR).

Upper outlier fence

Q3 + 1.5(IQR).

IQR outlier rule

An observation below the lower fence or above the upper fence is considered an outlier.

Does a value above Q3 automatically count as an outlier?

No. It must be above Q3 + 1.5(IQR).

Does a value below Q1 automatically count as an outlier?

No. It must be below Q1 − 1.5(IQR).

What should you do when you find an outlier?

Investigate it; correct errors if appropriate, understand its cause, and consider analyses with and without it rather than automatically deleting it.

Boxplot

A display of a quantitative distribution based on quartiles and useful for showing center, spread, skewness, and possible outliers.

Side-by-side boxplots

Used to compare a quantitative variable across categories of a categorical variable.

Side-by-side boxplot variable combination

One quantitative variable and one categorical variable.

Comparing boxplot centers

Compare the medians.

Comparing boxplot spread

Compare IQRs and overall ranges as appropriate.

Larger IQR

Indicates greater variability in the middle 50% of the observations.

Can a larger IQR prove there are more outliers?

No.

Contingency table

A table displaying counts for combinations of two categorical variables.

Contingency table variable combination

Two categorical variables.

Marginal distribution

The distribution of one categorical variable found from the totals in the margins of a contingency table.

Conditional distribution

The distribution of one variable restricted to individuals satisfying a condition on another variable.

Conditional percentage rule

The group named after words such as "among," "of," or "given" determines the denominator.

"Of customers who renewed" denominator

Use the total number of customers who renewed.

"Among Premium customers" denominator

Use the total number of Premium customers.

Independence

Two categorical variables are independent when the conditional distribution of one is the same or roughly the same across categories of the other.

Association

Two categorical variables are associated when the conditional distributions differ across categories.

How to test association from a contingency table

Compare appropriate conditional percentages, not just raw counts.

Same conditional percentages

Indicates the variables appear independent.

Different conditional percentages

Indicates the variables appear associated.

Why can raw counts be misleading in contingency tables?

Groups may have different sizes, so conditional percentages are needed for a fair comparison.

Mosaic plot

A graphical display used to examine the relationship between two categorical variables.