Source: MATH 399N Statistics Discussions, Weeks 1–2 | Arizona State University
Tags: mean, median, mode, range, standard deviation, central tendency, measures of variation, misleading graphs, pie chart errors, truncated axis, statistics, MATH 399N
Difficulty: Introductory | Prerequisites: Basic arithmetic, familiarity with percentages
This is where most introductory statistics courses begin: learning to describe a dataset with a handful of numbers rather than staring at a wall of raw values. Central tendency (mean, median, mode) tells you where the data clusters, and measures of variation (range, standard deviation) tell you how spread out it is. Week 2 shifts to reading graphs critically, because a chart that looks right can still mislead if labels are missing, axes are truncated, or slices do not add to 100%. You need arithmetic and a basic sense of percentages; nothing more advanced yet.
Mean, median, and mode each summarise a dataset's centre in a different way. Range and standard deviation measure how spread out the values are. Misleading graphs, such as pie charts with missing slices or bar charts with truncated axes, can distort a reader's perception of the data, so always check labels, totals, and baselines before drawing conclusions.
Mean (arithmetic average)
The sum of all values divided by the number of values. In simple terms, this is the "average" most people think of first.
Median
The middle value when all observations are arranged from smallest to largest. If there is an even number of values, take the average of the two middle ones. Think of it as the value that splits the dataset in half.
Mode
The value that appears most frequently in a dataset. A set can have no mode, one mode, or several modes. In simple terms, it is the most popular value.
Range
The difference between the largest and smallest values in the dataset (max minus min). It gives a rough sense of spread but is sensitive to outliers.
Standard deviation
A measure of how far, on average, each data point sits from the mean. A small standard deviation means the values cluster tightly around the mean; a large one means they are spread out. Think of it as the typical distance from the centre.
Measures of central tendency
Statistics that describe the centre or typical value of a dataset: mean, median, and mode.
Measures of variation (measures of spread / dispersion)
Statistics that describe how spread out the data are: range, standard deviation, variance, and interquartile range.
Misleading graph
A chart or plot that distorts the viewer's perception of the data, whether through missing labels, a truncated axis, disproportionate slices, or omitted categories.
Truncated axis (baseline manipulation)
A graph whose vertical axis does not start at zero, making small differences look dramatically large. Also called "omitting the baseline."
Add every value in the dataset, then divide by the number of values.
Example: the top 10 U.S. causes of death (2016 mortality-rate percentages) sum to 74%. Dividing by 10 gives a mean of 7.4%.
The mean is sensitive to extreme values. One very large or very small observation can pull it noticeably higher or lower.
Sort the data from smallest to largest.
If the count of values is odd, the median is the single middle value. If the count is even, average the two middle values.
Example: with 10 mortality-rate percentages sorted in order, the median falls between the 5th and 6th values. For both 2016 and 2017 data, the median was 5.2%.
The median is resistant to outliers, so it is often more representative than the mean when data are skewed.
The mode is whichever value appears most often.
A dataset may have no mode (all values unique), one mode, or multiple modes.
Example: in the mortality-rate dataset, the only repeated value across both years was 1.8% (kidney disease), making it the mode.
Subtract the smallest value from the largest: Range = Max − Min.
Example: 2016 mortality rates ranged from 1.6% (suicide) to 23.1% (heart disease), giving a range of 21.5 percentage points.
The range is easy to compute but tells you nothing about how the values in between are distributed.
Subtract the mean from each value to get the deviations.
Square each deviation, sum the squares, then divide by the number of values (population SD) or by n − 1 (sample SD).
Take the square root of that result.
Example: the standard deviation of the 2016 mortality-rate percentages was approximately 2.2%, meaning individual causes of death typically deviated about 2.2 percentage points from the mean of 7.4%.
A small standard deviation indicates the data points cluster near the mean; a large one indicates wide spread.
Every slice of a pie chart should be labelled, and all slices must add to 100%.
If you add up the labelled percentages and the total is less than 100%, some data is missing or hidden. Example: a pie chart of San Francisco real-estate purchases by industry totalled only 91.1% when the labelled slices were summed, leaving 8.9% unaccounted for.
Too many tiny slices make a pie chart dense and hard to read. Consider grouping small categories into an "Other" slice, or switching to a bar chart.
When a bar or line chart's vertical axis starts at a value above zero, small differences between bars look exaggerated.
Always check where the y-axis begins. A change from 98 to 100 looks enormous on a truncated axis but is a 2% shift on a full-scale one.
This technique is sometimes used deliberately to dramatise small changes in news graphics and marketing materials.
Check that all segments or slices add to 100% (for pie charts) or that the baseline is clearly shown.
Verify every segment is labelled with its value.
Ask whether the visual impression matches the numerical reality. If the chart makes a difference look huge, check the scale.
Consider whether a different chart type would communicate the data more clearly.
\text{Mean} = \bar{x} = \frac{\sum x_i}{n}\text{Median position} = \frac{n+1}{2}\text{Range} = x_{\max} - x_{\min}\text{Population standard deviation} = \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}\text{Sample standard deviation} = s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}Note the denominator difference: population SD divides by N (the full count); sample SD divides by n − 1 (Bessel's correction) to avoid underestimating the true spread.
Public health agencies (such as the CDC) use mean mortality rates across causes of death to identify where resources should be directed. If heart disease accounts for 23% of deaths while kidney disease accounts for 1.8%, the mean and range immediately highlight how concentrated the burden is. Journalists and policymakers rely on these same summary statistics to spot trends year over year, for instance noticing that accidental deaths rose from 5.9% to 6.0% between 2016 and 2017.
Students often think the mean is always the best measure of centre. It is not. When a dataset is heavily skewed or contains outliers, the median is usually more representative.
Students sometimes report "no mode" when they see only one repeated value. A single repeated value is still a mode; "no mode" applies only when every value is unique.
A common error is dividing by n instead of n − 1 when computing a sample standard deviation. Use n − 1 for samples (Bessel's correction) and N for full populations.
Students frequently assume a pie chart is accurate just because it looks professional. Always add up the labelled percentages to verify they reach 100%.
⚠️ You will almost certainly be asked to compute the mean, median, mode, range, and standard deviation from a small dataset and then interpret each value in context.
⚠️ Know the difference between population SD (divide by N) and sample SD (divide by n − 1). Exams frequently test whether you use the right denominator.
⚠️ Be prepared to identify what makes a graph misleading: missing labels, slices not summing to 100%, truncated baselines, and disproportionate visual elements.
⚠️ "Interpret in context" means writing a sentence that connects the number to the real-world meaning, not just stating the value. For example: "The mean mortality rate of 7.4% tells us that, on average, each of the 10 leading causes accounted for about 7.4% of all deaths."
True or False: The median is always equal to the mean. Answer: False. They are equal only in a perfectly symmetrical distribution.
Fill in the blank: The range is calculated by subtracting the ______ value from the ______ value. Answer: smallest, largest.
True or False: A pie chart whose labelled slices add to 85% is presenting all the data. Answer: False. 15% of the data is unlabelled or missing.
Fill in the blank: When computing the standard deviation of a sample, divide by ______ instead of N. Answer: n − 1.
True or False: A dataset where every value is unique has no mode. Answer: True.
Q: Given the values 5.9, 5.6, 5.2, 4.2, 2.9, 1.9, 1.8, 1.6, 23.1, 21.8, calculate the mean and interpret it.
A: Sum = 74.0. Mean = 74.0 / 10 = 7.4. On average, each of the 10 leading causes of death accounted for about 7.4% of all U.S. deaths in that year.
Q: Using the same dataset, what is the median and why might it be more useful than the mean here?
A: Sorted values: 1.6, 1.8, 1.9, 2.9, 4.2, 5.2, 5.6, 5.9, 21.8, 23.1. The median is (4.2 + 5.2) / 2 = 4.7 (or, as reported in the source, 5.2 depending on the exact midpoint convention used). The median is more useful because the two largest values (heart disease and cancer) are far above the rest, pulling the mean upward.
Q: A pie chart's labelled slices add up to 91.1%. What should you conclude?
A: 8.9% of the data is unaccounted for. The chart is misleading because it does not represent the full dataset. The missing portion could change the reader's interpretation of relative proportions.
Q: A bar chart comparing two products' sales starts its y-axis at 950 units instead of zero. Product A sold 980 units and Product B sold 1,000 units. What is the problem?
A: The truncated axis makes the 2% difference look enormous. On a full scale starting at zero, the bars would appear nearly identical. The chart exaggerates the gap between the two products.
Q: Explain the difference between population standard deviation and sample standard deviation.
A: Population SD divides the sum of squared deviations by N (the total number in the population). Sample SD divides by n − 1 (Bessel's correction) to account for the fact that a sample tends to underestimate the true variability of the population.
Central tendency and variation are the foundation for nearly everything that follows in this course. The z-score (Week 5) directly uses the mean and standard deviation to express how far a value is from centre. Confidence intervals (Week 6) rely on the standard deviation to calculate margins of error. Hypothesis testing (Week 7) compares sample means and uses pooled standard deviations. Being comfortable with these basics makes every later topic easier to follow.
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