Difficulty: Beginner to Intermediate | Prerequisites: Foundations of Statistics notes (variable types)
Once you have collected data, you need to make sense of it. This topic covers the tools for summarising and visualising both categorical and quantitative data, plus the measures (mean, median, standard deviation, correlation) that quantify patterns. These are the workhorses of descriptive statistics and the building blocks for every inference method that follows.
Categorical data is summarised with frequency tables and bar charts. Quantitative data is summarised with measures of centre (mean, median), spread (range, IQR, standard deviation), and visualised with histograms and boxplots. The correlation coefficient r measures the strength and direction of a linear relationship between two quantitative variables, but it says nothing about causation.
Frequency table
A table listing each category (or interval) and the count of observations in it.
Relative frequency table
A frequency table that shows proportions or percentages instead of raw counts.
Bar graph
A chart using bars to display counts or proportions for categorical data. Bars do not touch.
Pie chart
A circle divided into slices representing proportions of a whole.
Contingency table (two-way table)
A table that cross-tabulates two categorical variables, showing joint frequencies.
Think of it as: a grid where one variable defines the rows and the other defines the columns.
Marginal totals
The row totals and column totals of a contingency table.
Conditional relative frequency
The proportion within a specific row or column of a contingency table. Useful for spotting relationships between two categorical variables.
Dotplot
A graph placing a dot for each observation along a number line. Good for small datasets.
Histogram
A graph using adjacent bars to show the frequency distribution of a quantitative variable over intervals (bins). Bars touch.
Boxplot (box-and-whisker plot)
A graph summarising a dataset through its five-number summary: minimum, Q1, median, Q3, maximum. Outliers are shown as individual points.
Mean (average)
The sum of all values divided by the number of values. Sensitive to outliers.
In simple terms, this means: the "balance point" of the data.
Median
The middle value when data are sorted. Resistant to outliers.
Think of it as: the value that splits the dataset in half.
Mode
The most frequently occurring value in a dataset.
Range
Maximum minus minimum. Sensitive to outliers.
Interquartile range (IQR)
Q3 minus Q1. Measures the spread of the middle 50% of the data. Resistant to outliers.
Standard deviation
The typical distance of data points from the mean. Sensitive to outliers.
Think of it as: a measure of how spread out the data is around its average.
Empirical rule (68-95-99.7 rule)
For bell-shaped, symmetric distributions: about 68% of data falls within 1 standard deviation of the mean, about 95% within 2, and nearly all within 3.
Z-score (standardised score)
The number of standard deviations a data point lies from the mean. Calculated as z = (x - µ) / σ.
Think of it as: a way to compare values from different distributions on a common scale.
Percentile
The value below which a given percentage of observations fall.
Quartile
Special percentiles that divide data into four equal parts: Q1 (25th), Q2/median (50th), Q3 (75th).
Five-number summary
Minimum, Q1, median, Q3, maximum. The basis for a boxplot.
Scatterplot
A graph plotting pairs of quantitative values as points on two axes. Used to visualise relationships.
Correlation coefficient (r)
A number between -1 and 1 measuring the strength and direction of the linear relationship between two quantitative variables.
Think of it as: how tightly the points on a scatterplot cluster around a straight line, and which way the line slopes.
Frequency tables count how many observations fall in each category.
Relative frequency tables convert those counts to proportions or percentages.
Bar graphs display counts or proportions as bars (bars do not touch, because the categories are distinct).
Pie charts show proportions as slices of a circle. Useful when you want to emphasise parts of a whole.
For two categorical variables, use a contingency table (two-way table). It shows joint frequencies, with marginal totals along the edges. Conditional relative frequencies (proportions within a row or column) reveal relationships between the variables.
Visualisation options for two categorical variables include side-by-side bar charts (comparing groups) and stacked bar charts (showing proportions within groups).
Describe a quantitative distribution by its shape, centre, spread, and outliers.
Shape:
Symmetric: left and right sides are roughly mirror images.
Skewed left: tail extends to the left.
Skewed right: tail extends to the right.
Visualisation tools:
Dotplots: one dot per observation on a number line. Best for small datasets.
Histograms: bars over intervals showing frequency. Bars touch because the variable is continuous.
Boxplots: built from the five-number summary. Show centre, spread, and outliers at a glance.
Mean (x̄): sum of values divided by count. Pulled toward outliers and skew.
Median: middle value when sorted. Resistant to outliers.
Mode: most frequent value.
In a symmetric distribution, mean ≈ median. In a skewed distribution, the mean is pulled toward the tail.
Outliers are extreme data points. They distort the mean and standard deviation but have little effect on the median and IQR. When outliers are present or data is skewed, the median and IQR are more reliable summaries than the mean and standard deviation.
Range: maximum minus minimum. Simple but sensitive to outliers.
Interquartile range (IQR): Q3 - Q1. Covers the middle 50% of the data. Resistant to outliers.
Standard deviation (s or σ): the typical distance from the mean. Sensitive to outliers.
For bell-shaped (roughly normal), symmetric distributions:
About 68% of data falls within 1 standard deviation of the mean.
About 95% within 2 standard deviations.
Nearly 100% within 3 standard deviations.
This is useful for quick estimates. For instance, if pulse rates have mean 72 and standard deviation 8, then roughly 95% of people have a pulse between 56 and 88.
A z-score tells you how many standard deviations a value sits from the mean.
Positive z: above the mean.
Negative z: below the mean.
z = 0: at the mean.
Z-scores allow comparison across different scales. A student scoring z = 1.5 in maths and z = 0.8 in English did relatively better in maths, even if the raw scores are not directly comparable.
A percentile is the value below which a given percentage of observations fall.
Quartiles split the data into four equal parts: Q1 (25th percentile), Q2 (50th, the median), Q3 (75th).
To calculate: sort the data, then find the median of the lower half (Q1) and upper half (Q3).
Minimum, Q1, Median, Q3, Maximum. This is the foundation of a boxplot and gives a quick picture of centre, spread, and symmetry.
Scatterplots display pairs of quantitative values. When reading one, assess:
Form: linear or nonlinear?
Direction: positive (both increase together) or negative (one increases as the other decreases)?
Strength: tight cluster or widely scattered?
Outliers: points far from the general pattern.
r measures the strength and direction of the linear relationship between two quantitative variables.
r = 1: perfect positive linear relationship.
r = -1: perfect negative linear relationship.
r = 0: no linear relationship.
Computed as the average of the products of the standardised scores of x and y.
Key properties of r:
Unitless: unaffected by changes in units of measurement.
Symmetric: the correlation between X and Y equals the correlation between Y and X.
Sensitive to outliers: a single extreme point can inflate or deflate r.
Does not imply causation: a strong correlation does not mean one variable causes changes in the other.
Only measures linear relationships: a strong curve can have r near 0.
Formula | What it calculates |
|---|---|
x̄ = Σx / n | Sample mean |
z = (x - µ) / σ | Z-score (population) |
z = (x - x̄) / s | Z-score (sample) |
IQR = Q3 - Q1 | Interquartile range |
Range = Max - Min | Range |
Empirical rule thresholds:
68% of data: µ ± 1σ
95% of data: µ ± 2σ
99.7% of data: µ ± 3σ
Hospitals use the empirical rule to flag abnormal lab results: a value more than 2 standard deviations from the mean triggers further investigation. Credit scoring models rely on z-scores to compare applicants across different income distributions. Weather forecasters report percentiles ("today's high is in the 90th percentile for this date") to contextualise temperatures. Scatterplots and correlation drive early analysis in fields from epidemiology (does exposure correlate with illness rates?) to finance (do two stocks move together?).
Students often report the mean for skewed data or data with outliers, when the median would be more appropriate. If the distribution is skewed or contains extreme values, use the median and IQR.
Students confuse histograms with bar charts. Histogram bars touch (continuous data, order matters). Bar chart bars do not touch (categorical data, order is arbitrary).
Students assume r = 0 means "no relationship." It means no linear relationship. The variables could have a strong curved association.
Students apply the empirical rule to skewed distributions. It only works for bell-shaped, approximately symmetric distributions.
Students think a high correlation (e.g. r = 0.95) proves causation. It does not. Correlation measures linear association, not cause and effect.
⚠️ Know when to use mean vs median. If the question mentions skew or outliers, the median is the better measure of centre.
⚠️ Be able to calculate the five-number summary and IQR from a small dataset. This is a common computation question.
⚠️ The empirical rule is frequently tested. Given a mean and standard deviation, expect to be asked for the interval containing 68% or 95% of the data.
⚠️ Know how to interpret r. Expect questions that give you a scatterplot or an r value and ask about the strength, direction, or form of the relationship.
⚠️ Remember that r only measures linear relationships and does not imply causation. Both points are heavily tested.
True or false: The median is more sensitive to outliers than the mean. (False. The mean is more sensitive. The median is resistant.)
Fill in the blank: For a bell-shaped distribution, about ____% of data falls within 2 standard deviations of the mean. (95%.)
True or false: A correlation coefficient of r = -0.9 indicates a weak relationship. (False. It indicates a strong negative linear relationship.)
Fill in the blank: The IQR is calculated as ______ minus ______. (Q3 minus Q1.)
True or false: A histogram's bars should have gaps between them. (False. Histogram bars touch because the data is continuous.)
Q: A dataset of home prices is right-skewed. Should you report the mean or the median as the typical price? Why?
A: The median. In a right-skewed distribution, a few very expensive homes pull the mean upward, making it unrepresentative of the typical home. The median is resistant to these outliers.
Q: Pulse rates in a class have a mean of 72 and a standard deviation of 8. Using the empirical rule, what interval contains about 95% of the pulse rates?
A: 72 ± 2(8) = 56 to 88. About 95% of pulse rates fall between 56 and 88 bpm.
Q: A student scores 80 on a test with mean 70 and standard deviation 5. What is the student's z-score, and what does it mean?
A: z = (80 - 70) / 5 = 2.0. The student scored 2 standard deviations above the mean.
Q: Two variables have a correlation of r = 0.02. Does this mean there is no relationship between them?
A: Not necessarily. r = 0.02 means there is essentially no linear relationship, but the variables could still have a strong nonlinear (curved) relationship. Always check the scatterplot.
Q: Given the dataset 3, 7, 8, 12, 15, 18, 22, find the five-number summary.
A: Minimum = 3, Q1 = 7, Median = 12, Q3 = 18, Maximum = 22. (With 7 values, the median is the 4th value. Q1 is the median of the lower half {3, 7, 8} = 7. Q3 is the median of the upper half {15, 18, 22} = 18.)
The summary measures here (mean, standard deviation, z-scores) are the raw ingredients for the sampling distributions, confidence intervals, and hypothesis tests covered later in STAT101. Scatterplots and correlation lead into linear regression, where you fit a line and use it for prediction. The distinction between resistant and non-resistant measures connects back to your choice of summary when outliers are present.
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