Course: PSY 301 Understanding Statistics, East Texas A&M Source: Moore & Notz, Statistics: Concepts and Controversies, 10th ed., Chapters 10, 11, 12 Covers: Exam 2 material
Difficulty: Introductory to Intermediate Prerequisites: Chapter 1 (types of variables), Chapter 3 (samples), Chapter 8 (measurement quality). You need to know the difference between categorical and quantitative variables before any of this will click.
Once you have collected data, the next step is to make sense of it. Chapters 10 and 11 cover how to display data visually, choosing the right graph for the right variable type. Chapter 12 moves from pictures to numbers, introducing the summary statistics (mean, median, standard deviation, and so on) that compress a dataset into a few useful values. Together, these chapters form the "descriptive statistics" toolkit. Everything you learn here feeds directly into the inferential methods in later chapters, where you draw conclusions beyond your sample.
Categorical data get bar graphs and pie charts. Quantitative data get histograms, stemplots, and boxplots. When summarising quantitative data numerically, the mean and standard deviation describe the centre and spread of symmetric distributions, while the median and interquartile range are better for skewed data or data with outliers.
Distribution
The pattern of values a variable takes, including which values occur and how often. Think of it as: the shape of your data when you graph it.
Bar graph
A graph that uses separated bars to display the count or percentage of individuals in each category. In simple terms, one bar per category, heights show frequency. Bars do not touch because categories are not continuous.
Pie chart
A circle divided into slices, where each slice represents a category's share of the whole. Think of it as: a visual way to show "parts of a total." Only sensible when the categories represent all possible options and their percentages must add to 100%.
Histogram
A graph of the distribution of a quantitative variable, using bars whose heights represent counts or percentages in each class interval. In simple terms, it looks like a bar graph, but the bars touch because the data are continuous. The x-axis is a number line.
Stemplot (stem-and-leaf plot)
A display that splits each data value into a "stem" (leading digits) and a "leaf" (trailing digit), then lines up the leaves beside their stems. Think of it as: a quick, hand-drawn histogram that preserves the actual data values.
Boxplot (box-and-whisker plot)
A graph based on the five-number summary that shows the median, quartiles, and potential outliers. In simple terms, the box spans the middle 50% of the data, the line inside the box is the median, and the "whiskers" extend to the smallest and largest observations that are not outliers.
Shape of a distribution
Described by symmetry (or lack of it), number of peaks, and the presence of outliers. Think of it as: when you look at a histogram, is it roughly a mirror image (symmetric), does it have a long tail to the right (right-skewed) or left (left-skewed)?
Symmetric distribution
A distribution where the left and right sides are roughly mirror images. In simple terms, if you fold the histogram in half, the two sides match up.
Skewed distribution
A distribution with a long tail on one side. Right-skewed (positive skew) has a tail stretching toward larger values. Left-skewed (negative skew) has a tail stretching toward smaller values. Think of it as: the tail points in the direction of the skew.
Mean (arithmetic average)
The sum of all values divided by the number of values. In simple terms, add everything up and divide by how many there are. The mean is pulled toward extreme values.
Median
The midpoint of a distribution, the value such that half the observations fall below it and half above. Think of it as: line up all the numbers in order and pick the one in the middle. If there is an even count, average the two middle values.
Mode
The value that occurs most frequently in a dataset. In simple terms, the most popular answer.
Range
The difference between the largest and smallest values. Think of it as: the simplest measure of spread, but very sensitive to outliers.
Quartiles (Q1 and Q3)
The values that divide the ordered data into four equal parts. Q1 is the median of the lower half; Q3 is the median of the upper half. In simple terms, Q1 marks the 25th percentile and Q3 marks the 75th percentile.
Interquartile range (IQR)
The difference Q3 minus Q1, representing the spread of the middle 50% of the data. Think of it as: a measure of spread that ignores the extremes.
Five-number summary
Minimum, Q1, Median, Q3, Maximum. In simple terms, five values that give you a quick snapshot of the data's centre, spread, and range.
Outlier
An observation that falls far outside the overall pattern of the data. Think of it as: a data point that does not fit. A common rule of thumb is that a value more than 1.5 times the IQR below Q1 or above Q3 is a suspected outlier.
Standard deviation
A measure of spread that describes how far values typically fall from the mean. In simple terms, a small standard deviation means data cluster tightly around the mean; a large one means they are more spread out.
Variance
The average of the squared deviations from the mean (using n minus 1 for a sample). Think of it as: the standard deviation before you take the square root. It is in squared units, which is why the standard deviation is more interpretable.
Resistant measure
A summary statistic that is not strongly affected by outliers or extreme values. The median and IQR are resistant. The mean and standard deviation are not. In simple terms, resistant measures hold steady even when a few wild values appear.
Choosing the right graph depends on the variable type:
Categorical variables: bar graph or pie chart.
Quantitative variables: histogram, stemplot, or boxplot.
When examining a distribution (histogram or stemplot), describe three things:
Shape: symmetric, left-skewed, right-skewed, unimodal, bimodal.
Centre: where the middle of the data falls (rough estimate from the graph).
Spread: how variable the data are, from minimum to maximum or by IQR.
Also note any outliers or unusual features.
Stemplots preserve individual data values, which histograms do not. They work well for small datasets (roughly under 50 values).
Boxplots are ideal for comparing distributions across groups side by side. They highlight the five-number summary visually.
Graphs can mislead:
Truncated y-axes exaggerate small differences.
Pictographs that scale area rather than height distort comparisons.
Using three-dimensional effects adds no information and obscures the data.
Always start bar graph y-axes at zero.
Measures of centre:
The mean is the balance point of the distribution. Sensitive to outliers and skew.
The median is the positional middle. Resistant to outliers.
For symmetric data without outliers, mean and median are close together; use the mean.
For skewed data or data with outliers, the median is the better summary.
Rule of thumb: if the distribution is right-skewed, the mean is pulled above the median. If left-skewed, the mean is pulled below the median.
Measures of spread:
Range is simple but unreliable, since it depends entirely on the two most extreme values.
IQR captures the spread of the middle 50% and is resistant to outliers.
Standard deviation measures typical distance from the mean. It is the most commonly used measure of spread for symmetric, roughly normal data.
Standard deviation is zero only when all values are identical. It can never be negative.
Standard deviation and mean go together. Median and IQR go together. Mixing them (e.g., reporting the mean with the IQR) is poor practice.
The five-number summary and the boxplot are a matched pair: the boxplot is simply the five-number summary drawn as a picture.
Choosing the right summary:
Symmetric, no outliers: report mean and standard deviation.
Skewed or outliers present: report median and IQR (with the five-number summary).
Mean:
x̄ = (sum of all values) / n
Where n is the number of observations.
Standard deviation (sample):
s = √[ Σ(xᵢ − x̄)² / (n − 1) ]
Steps: subtract the mean from each value, square each result, sum the squares, divide by n minus 1, take the square root.
Interquartile range:
IQR = Q3 − Q1
Outlier rule of thumb:
A value is a suspected outlier if it falls below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR).
Income data are almost always right-skewed (a small number of very high earners pull the tail out), which is why economists typically report median household income rather than mean household income. Reporting the mean would give a misleadingly high picture of what a typical household earns.
Boxplots are widely used in medical and psychological research to compare groups at a glance, for instance comparing anxiety scores across treatment conditions, because they simultaneously show centre, spread, and potential outliers.
Students often describe any histogram that is not perfectly flat as "skewed." Skew refers specifically to a long tail on one side, not to any departure from uniformity.
Students sometimes think the median always sits inside the box of a boxplot. It does, by definition, the line inside the box is the median.
Students frequently confuse the standard deviation with the range. The standard deviation is a typical distance from the mean, not the total distance from smallest to largest.
Students forget that the mean and standard deviation should only be reported together, and the median and IQR should only be reported together. Mixing pairs is a common mistake on exams.
⚠️ Be able to match graph types to variable types. Expect questions like "which graph is appropriate for this dataset?"
⚠️ When presented with a histogram, describe shape, centre, spread, and outliers. Leaving any of these out costs marks.
⚠️ Know when to use mean/standard deviation vs. median/IQR. The deciding factor is skewness and outliers.
⚠️ Be comfortable calculating mean, median, Q1, Q3, IQR, and standard deviation by hand. A calculator that does square roots is sufficient.
⚠️ The outlier rule (1.5 times IQR) is a common exam question. Know how to apply it.
⚠️ Exams are cumulative. Material from Chapters 1, 3, and 8 may reappear alongside these topics.
True or false: A histogram is the appropriate graph for a categorical variable.
Fill in the blank: The __________ is a resistant measure of centre.
True or false: If a distribution is right-skewed, the mean is typically greater than the median.
Fill in the blank: Standard deviation can never be __________.
True or false: The IQR measures the spread of all the data.
Answers: 1. False (use a bar graph). 2. Median. 3. True. 4. Negative. 5. False (it measures the spread of the middle 50%).
Q: A dataset of exam scores has a five-number summary of 42, 67, 78, 85, 99. Calculate the IQR and determine whether a score of 38 would be considered an outlier.
A: IQR = Q3 − Q1 = 85 − 67 = 18. The lower fence is Q1 − 1.5(IQR) = 67 − 27 = 40. Since 38 is below 40, it would be considered a suspected outlier.
Q: Why is the median a better measure of centre than the mean for household income data?
A: Income distributions are strongly right-skewed. A small number of very high incomes pull the mean upward, making it unrepresentative of a typical household. The median is resistant to these extreme values and better reflects the centre of the distribution.
Q: A researcher presents a bar graph of favourite ice cream flavours. The bars are touching. What is wrong with this graph?
A: Bars in a bar graph should be separated by gaps to indicate that the categories are distinct and not continuous. Touching bars imply a continuous quantitative variable (as in a histogram), which is misleading for categorical data.
Q: You are told a dataset is symmetric and has no outliers. Which pair of summary statistics should you report?
A: The mean and standard deviation, because they use all the data and are most informative when the distribution is symmetric without extreme values.
Q: What does it mean for a measure to be "resistant"?
A: A resistant measure is one that does not change much when a few extreme values (outliers) are added or removed. The median and IQR are resistant; the mean and standard deviation are not.
These chapters connect directly to Chapter 13 (normal distributions). The mean and standard deviation are the two parameters that define a normal distribution, so you need to be comfortable with them before moving on. Boxplots and the concept of outliers also reappear when interpreting results from hypothesis tests in Chapters 21 and 22.
distribution, bar graph, pie chart, histogram, stemplot, stem-and-leaf, boxplot, box-and-whisker, shape, symmetric, skewed, right-skewed, left-skewed, unimodal, bimodal, mean, average, median, mode, range, quartile, Q1, Q3, interquartile range, IQR, five-number summary, outlier, 1.5 IQR rule, standard deviation, variance, resistant measure, descriptive statistics, PSY 301, Moore and Notz, understanding statistics