Source: Moore & Notz, Statistics: Concepts and Controversies (10th ed.), Chs. 1–10
Tags: descriptive statistics, variables, categorical, quantitative, mean, median, mode, standard deviation, variance, histogram, boxplot, bar chart, data visualisation, STAT 101, intro statistics
Difficulty: Beginner | Prerequisites: None. This is the starting point for the course.
Descriptive statistics is the half of the discipline concerned with organising, summarising, and displaying data you already have. Before you can test a hypothesis or make predictions, you need to know what your data looks like: what type it is, where it centres, how spread out it is, and whether anything unusual is going on. This material forms the foundation for everything else in STAT 101 and comes up in every exam. If you skipped the first few weeks, start here.
Descriptive statistics gives you tools to summarise a dataset with a few numbers (centre, spread) and pictures (graphs). The first step is always identifying what kind of variable you have, because that determines which summaries and visuals are appropriate.
Variable
A characteristic that differs from one individual or observation to the next. Height, eye colour, and test score are all variables. In simple terms, a variable is anything you measure or record about each subject in your study.
Categorical variable (qualitative variable)
A variable whose values are group labels rather than numbers. Examples: blood type, favourite colour, yes/no responses. Think of it as a variable that sorts individuals into buckets.
Quantitative variable (numerical variable)
A variable whose values are meaningful numbers on which arithmetic makes sense. Examples: weight in kg, income in dollars, temperature. In simple terms, if you can sensibly average it, it is quantitative.
Discrete variable
A quantitative variable that can only take isolated values, usually whole numbers. Number of siblings, number of courses enrolled. Think of it as a variable you count rather than measure.
Continuous variable
A quantitative variable that can take any value within some interval. Height, time, temperature. Think of it as a variable you measure rather than count.
Distribution
The pattern of values a variable takes and how often each value occurs. In simple terms, a distribution answers "what values showed up, and how common was each one?"
Mean (arithmetic mean, average)
The sum of all observations divided by the number of observations. This is the balance point of the data. In simple terms, it is the "fair share" value if you spread the total evenly.
Median
The middle value when all observations are arranged in order. If there is an even number of observations, the median is the average of the two middle values. Think of it as the value that splits the dataset in half.
Mode
The value that appears most frequently. A dataset can have one mode, more than one, or none. In simple terms, the mode is the most popular value.
Range
The difference between the largest and smallest values. Quick to compute but sensitive to outliers.
Interquartile range (IQR)
The difference between the third quartile (Q3) and the first quartile (Q1). It captures the spread of the middle 50% of the data. Think of it as the range that ignores the extremes.
Variance
The average of the squared deviations from the mean. It quantifies how far values typically sit from the centre, in squared units. In simple terms, variance measures how spread out the data is, but in awkward units (e.g. "square dollars").
Standard deviation (SD)
The square root of the variance. It is expressed in the same units as the original data and is the most commonly reported measure of spread. Think of it as a typical distance from the mean.
Outlier
An observation that falls well outside the overall pattern of the data. A common rule of thumb: any value more than 1.5 × IQR below Q1 or above Q3 is flagged as an outlier.
Five-number summary
The minimum, Q1, median, Q3, and maximum. These five values give a compact picture of centre, spread, and range. A boxplot is the visual version of this summary.
Skewness
A description of how asymmetric a distribution is. Right-skewed (positive skew) means a long tail to the right; left-skewed (negative skew) means a long tail to the left. In simple terms, the skew tells you which direction the tail stretches.
The first question with any dataset: is each variable categorical or quantitative?
Categorical variables get summarised with counts and proportions. Quantitative variables get summarised with means, medians, and standard deviations.
A common trap: numbers used as labels (e.g. jersey numbers, postcodes) are categorical, not quantitative. If averaging the values would be meaningless, the variable is categorical regardless of whether it looks numerical.
Mean: best used when the distribution is roughly symmetric and free of extreme outliers. Pulled towards outliers.
Median: best used when the distribution is skewed or contains outliers. Resistant to extreme values.
Mode: useful for categorical data or for spotting peaks in a distribution. Less commonly used as a standalone summary for quantitative data.
Rule of thumb: if mean and median differ noticeably, the distribution is probably skewed.
Range: simple but unreliable; two very different datasets can share the same range.
IQR: a resistant measure of spread; pairs naturally with the median.
Standard deviation: the default measure of spread for roughly symmetric data; pairs naturally with the mean.
Variance: the squared version of SD; important in formulas but rarely reported on its own in applied work.
Symmetric distribution, no major outliers: report mean and standard deviation.
Skewed distribution or outliers present: report median and IQR.
This pairing is not optional style; the wrong combination gives a misleading picture.
Bar chart / bar graph: for categorical variables. Each category gets its own bar, and the height (or length) represents the count or proportion. Bars do not touch.
Pie chart: also for categorical data, but harder to read than bar charts. Generally avoid unless there are very few categories.
Histogram: for quantitative variables. Data is binned into intervals, and the bar height shows frequency or relative frequency. Bars touch because the variable is continuous.
Stemplot (stem-and-leaf plot): a quick way to see the shape of small datasets while preserving the actual data values.
Boxplot (box-and-whisker plot): a visual of the five-number summary. The box spans Q1 to Q3, the line inside is the median, and the whiskers extend to the smallest and largest non-outlier values. Outliers are plotted individually.
Scatterplot: for the relationship between two quantitative variables. One goes on the x-axis, one on the y-axis, and each observation is a dot.
For histograms, describe the shape (symmetric, left-skewed, right-skewed, bimodal), centre (roughly where the peak is), spread (how wide the data stretches), and any outliers.
For boxplots, compare the position of the median line within the box. If the median sits closer to Q1, the distribution is right-skewed; closer to Q3, left-skewed.
Mean
x̄ = (Σxᵢ) / n
where xᵢ is each observation and n is the total number of observations.
Variance (sample)
s² = Σ(xᵢ − x̄)² / (n − 1)
Note the denominator is n − 1, not n. This is called Bessel's correction and accounts for the fact that you are estimating from a sample rather than the whole population.
Standard deviation (sample)
s = √s²
IQR
IQR = Q3 − Q1
Outlier fences
Lower fence = Q1 − 1.5 × IQR Upper fence = Q3 + 1.5 × IQR
Any value below the lower fence or above the upper fence is flagged as a potential outlier.
Descriptive statistics show up every time someone reports an average salary, a median house price, or a graph of stock performance over time. News outlets prefer the median for income data precisely because income distributions are right-skewed, and the mean would overstate what a typical person earns. Understanding which summary to use is not just an exam skill; it protects you from being misled by statistics in everyday life.
Students often think the mean is always the best measure of centre. It is not. For skewed data, the median is more representative.
Students confuse histograms with bar charts. Histograms are for quantitative data with touching bars; bar charts are for categorical data with separated bars.
Students sometimes believe standard deviation can be negative. It cannot, because it is the square root of the variance, which is a sum of squared values.
Students forget that variance uses n − 1 in the denominator for a sample. Using n instead gives the population variance, which is a different formula for a different context.
⚠️ Expect questions asking you to pick the correct summary (mean vs. median) for a given distribution shape.
⚠️ You will almost certainly need to compute standard deviation or IQR by hand from a small dataset.
⚠️ Identifying variable types (categorical vs. quantitative, discrete vs. continuous) is tested repeatedly and is easy marks if you practise.
⚠️ Describing the shape of a histogram (symmetric, skewed, bimodal, outliers) is a staple of STAT 101 exams.
True or false: A postcode is a quantitative variable because it is made up of numbers.
Fill in the blank: For a right-skewed distribution, the mean is typically _______ than the median.
True or false: The IQR measures the spread of the entire dataset.
Fill in the blank: The standard deviation is the _______ of the variance.
True or false: In a histogram, the bars should not touch each other.
Answers: 1. False (postcodes are categorical labels). 2. Greater. 3. False (it measures the middle 50%). 4. Square root. 5. False (histogram bars touch; bar chart bars do not).
Q: A dataset of household incomes in a city is heavily right-skewed. Which measure of centre and which measure of spread should you report, and why?
A: Report the median and the IQR. Right-skewed data has a long upper tail that pulls the mean upward, making it unrepresentative of a typical household. The median resists that pull, and the IQR ignores the extreme values at both ends.
Q: You are given the five-number summary: Min = 12, Q1 = 25, Median = 30, Q3 = 40, Max = 95. Is the value 95 an outlier?
A: IQR = 40 − 25 = 15. Upper fence = 40 + 1.5(15) = 62.5. Since 95 > 62.5, yes, 95 is an outlier by the 1.5 × IQR rule.
Q: What is the difference between a histogram and a bar chart?
A: A histogram displays the distribution of a single quantitative variable using contiguous (touching) bars over intervals. A bar chart displays counts or proportions for a categorical variable with separated bars. The distinction matters because the appropriate graph depends on the variable type.
Q: Calculate the sample standard deviation of the dataset: 4, 8, 6, 5, 7.
A: Mean = (4 + 8 + 6 + 5 + 7) / 5 = 6. Squared deviations: 4, 4, 0, 1, 1. Sum = 10. Variance = 10 / (5 − 1) = 2.5. Standard deviation = √2.5 ≈ 1.58.
Q: A variable records whether a patient's test result is "positive," "negative," or "inconclusive." What type of variable is this?
A: Categorical (qualitative). The values are group labels, not numbers on which arithmetic is meaningful.
This material connects directly to normal distributions and z-scores (Document 2), because the mean and standard deviation are the two parameters that define a normal curve. It also underpins inferential statistics (Document 3): you cannot interpret a confidence interval or a hypothesis test without first understanding what the sample mean and standard deviation represent.
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