Chapter 2: Descriptive Statistics

What are the two main types of data?

Categorical (qualitative) and numerical (quantitative).

What is nominal data?

Categorical data whose categories have no natural ordering.

What is ordinal data?

Categorical data whose categories have a meaningful order.

Which averages make sense for nominal data?

Mode only; mean and median are not meaningful.

Which averages make sense for ordinal data?

Mode and median; the mean is generally not meaningful.

What is discrete numerical data?

Numerical data with separated possible values, often counts.

What is continuous numerical data?

Numerical data that can theoretically take any value in an interval.

What distinguishes ratio data from interval data?

Ratio data have a meaningful natural zero; interval data do not.

What five features should you examine when describing a distribution?

Location, spread, shape, outliers, and other unusual features (such as distinct groups or unexpectedly frequent values).

What does location describe?

The centre or typical value of a distribution.

What does spread describe?

How variable or dispersed the observations are.

What is positive skew?

A distribution with a long tail extending to the right.

What is negative skew?

A distribution with a long tail extending to the left.

What is the five-number summary?

Minimum, lower quartile, median, upper quartile and maximum.

What does the median represent?

The middle ordered value; a measure of location.

What are the lower and upper quartiles?

The lower quartile (qL) separates approximately the lowest 25% of observations from the rest; the upper quartile (qU) separates approximately the highest 25% from the rest.

How is the range calculated?

Maximum - minimum.

How is the interquartile range (IQR) calculated?

Upper quartile - lower quartile: qU - qL.

What are the range and IQR measures of?

Spread or variability.

What is the sample mean?

The arithmetic average: x̄ = (1/n)Σxi.

What does the sample variance measure?

A measure of spread based on squared deviations from the sample mean: s² = [Σᵢ₌₁ⁿ(xᵢ − x̄)²]/(n − 1).

What is the sample standard deviation?

The square root of sample variance; it has the data's original units.

When is median often preferred to mean?

When data are strongly skewed or contain outliers.

Why is the mean sensitive to outliers?

Extreme values directly affect the arithmetic average.

When is IQR often preferred to standard deviation?

For skewed data or data containing outliers.

What is the mode?

The value or category occurring most frequently.

For what data types is the mode especially useful?

Categorical and discrete numerical data.

When do ratios and percentage changes make sense?

Only when the variable is measured on a ratio scale.

Why can percentage temperature changes be misleading in Celsius?

0°C is arbitrary, so Celsius is an interval rather than ratio scale.

Give examples of ratio-scale variables.

Temperature in Kelvin, age, height and mass.

Give examples of interval-scale variables.

Temperature in Celsius or Fahrenheit, and calendar year.

What is the 100p% sample quantile using the STAT0002 method?

x_(p(n+1)), using the ordered observations.

What should you do if p(n+1) is not an integer?

Use linear interpolation between the two surrounding ordered observations.

How do you linearly interpolate a sample quantile?

Move the fractional part of p(n+1) from x_(i) towards x_(i+1).

What quantile type does STAT0002 use in R?

type = 6.

What quantile type does R's quantile() use by default?

type = 7.

Is comparing the mean and median a reliable test for skewness?

No. Mean and median can give a misleading indication of skewness.

What should you primarily use to judge the shape or skewness of data?

A suitable plot of the distribution.

How does standardised sample skewness react to extreme values?

It is sensitive to very large or very small observations.

How does quartile skewness react to extreme values?

It is much less sensitive because it is based on quartiles.

Can different measures of skewness disagree?

Yes. They can disagree with each other and with the appearance of a plot.

Why must you consider what numerical data represent before summarising them?

The measurement scale and context determine which summaries are meaningful.

Why is the ordinary mean of directions 10° and 350° inappropriate?

The values wrap around; both are close to 0°, so their meaningful centre is 0°.

What is a frequency distribution?

A table showing how many observations fall into each of several non-overlapping classes.

What is the modal class?

The class or interval with the highest frequency.

How is relative frequency calculated?

Frequency of a class divided by the total number of observations (n).

What is cumulative frequency?

The running total of frequencies up to and including a class.

What is cumulative relative frequency?

Cumulative frequency divided by n; the proportion of observations up to a class boundary.

What does the area of a histogram bar represent?

The relative frequency in that interval. In a normalised histogram, all bar areas sum to 1.

What is the height of a normalised histogram bar?

Relative frequency ÷ bin width = frequency ÷ (n × bin width).

Why must histogram bin widths be accounted for?

With unequal widths, frequency alone gives misleading bar heights; area must reflect the amount of data.

What distinguishes a barplot from a histogram?

A barplot displays categorical or discrete numerical data with separated bars; a histogram displays binned continuous numerical data.

Are outliers necessarily incorrect observations?

No. Extreme values can arise naturally; do not automatically remove them.

What are boxplots especially useful for?

Comparing distributions’ location, spread and shape; their main purpose is not outlier detection.

Why are pie charts often ineffective?

Angles and areas are hard to compare accurately, and many categories create clutter.

Why are multiple pie charts hard to compare?

Small changes in category proportions between charts are visually difficult to judge.

When should a time-series plot be used?

To display change over time; put time on the horizontal axis to reveal trends, cycles and unusual changes.

Why was the Buchanan vote in Palm Beach County statistically unusual?

The official 3,407 votes were far higher than the pattern across other Florida counties suggested.

Why compare vote percentages or similar counties rather than raw totals alone?

Counties differ in population size and characteristics, so raw vote totals are not directly comparable.

Why was Palm Beach County excluded when fitting the regression model?

Its extreme result could influence the fitted model, and the aim was to predict it independently from other counties.

What did the election model predict for Buchanan’s Palm Beach vote?

371 votes, with a 95% prediction interval of 219–534, versus the official 3,407.

What data transformations were used in the election model?

A square-root transformation of Buchanan’s vote percentage and log10 transformations of some explanatory variables.

What does one unit on a log10 scale mean?

A tenfold change on the original scale; 10, 100 and 1,000 are equally spaced on a log10 axis.

What does a scatterplot show?

The relationship between two paired numerical variables, with each point representing one pair of observations.

Which variables normally go on the x-axis and y-axis of a scatterplot?

x: explanatory or predictor variable; y: response or outcome variable.

What four features should you examine in a scatterplot?

Direction, form (linear or curved), strength of association and unusual points/outliers.

Why can an outlier appear in a scatterplot but not in separate one-variable plots?

A point may have ordinary individual values but an unusual combination of the two variables.

Why might we transform a dataset?

To make a distribution more symmetric or a relationship more linear so a simple statistical method’s assumptions fit better.

How can a logarithmic transformation affect positively skewed data?

It compresses large positive values relative to small ones, often reducing right skew.

Which transformations might reduce positive skew?

For positive data, try log(y) or a power y^c with c < 1, such as sqrt(y).

Which transformations might reduce negative skew?

For positive data, try powers y^c with c > 1, such as y² or y³, to stretch the upper tail.

When is taking logarithms directly invalid?

When observations are zero or negative; a suitable positive shift may sometimes be applied first.

Why might we straighten a curved scatterplot?

An approximately linear relationship is easier to interpret and better suited to methods such as linear regression.

Can you transform x, y or both to straighten a scatterplot?

Yes. Transform the explanatory variable, response variable or both, then inspect the new plot.

How could y = x² be straightened for x ≥ 0?

Plot sqrt(y) against x: then sqrt(y) = x, a straight-line relationship.

What does Pearson's sample correlation coefficient measure?

The strength of linear association between two variables.

What values can Pearson's sample correlation coefficient take?

From −1 to +1.

What does Spearman's rank correlation coefficient measure?

How close an association is to being monotone, using the product-moment correlation of paired ranks.

What is a monotonic association?

As x increases, y tends to increase or decrease; the relationship need not be linear.

Why might Spearman's correlation be preferred to Pearson's?

It is less sensitive to outliers and measures monotone association, including nonlinear relationships.

What is sample quartile skewness?

[(qU − m) − (m − qL)]/(qU − qL), where qL, m and qU are the sample quartiles.

What is an advantage of a stem-and-leaf plot?

It preserves the entire dataset (perhaps rounded) in sorted order, making the five-number summary easy to calculate.

What is a dotplot?

Each observation is a dot; repeated values (perhaps after rounding) are stacked vertically.

How are the whiskers of a standard boxplot determined?

They reach the most extreme observations within 1.5 IQR of the corresponding quartiles. More extreme values are shown individually.

What is an empirical cumulative distribution function (ECDF)?

It gives the cumulative relative frequency: the proportion of observations less than or equal to each value.

What is Tukey and Mosteller's bulging rule?

A guide that uses the curve's bulging direction to suggest transformations of x, y or both that may straighten a scatterplot.

What is the Box-Cox transformation?

BC(x) = (x^c - 1)/c when c ≠ 0, and BC(x) = ln(x) when c = 0. It is used to transform positive data, with c controlling the transformation.