Mean, median and mode
These are three ways to describe the centre of data. They answer different questions and react differently to unusual values.
The mean is sum divided by valid count. It uses every value and is the balance point: deviations from it sum to zero. The median is the middle sorted value, or the average of the middle two for an even-sized numeric list. It depends on position and is less sensitive to extreme magnitudes. The mode is the most frequent value or category; there may be several modes.
Mean and SD are often useful for a roughly symmetric, single-group numerical distribution. Median and IQR often describe a typical observation in a long-tailed distribution. For a category, report counts or proportions and possibly the mode. A total is related to the mean, not generally to the median.
W2 reports platform-use mean 1,592 and median 1,426 minutes per student over the term. The mean is pulled upward by high users. For a typical student, median plus IQR is natural. For total recorded workload, sum or valid n times the observed mean is appropriate. Multiplying an observed mean by a larger population count with missing values introduces an assumption.
Sources: W2 PDF pp. 2-60; L2 Sections 1-8. The personalised Lab 2 has 240 synthetic students, unlike the 1,200-student lecture dataset. Do not mix their numbers.
The mean and SD formulas in the lecture are applied to your eight generated scores. The code checks your working after you try it.
Mean, median and skewness are clues; the four plots show what they miss. Changing bins changes the picture, not the data.
The lecture's question-first approach decides which summary to use: a typical student and total observed workload are different requests.
Practice problems
Supplementary study questions on the lecture concepts — not verbatim exam questions. Each answer counts toward spaced review.
1. Conceptual understanding
Definitions and distinctions.
Which summary often describes typical use in a strongly right-tailed distribution?
Which quantity reconstructs total observed workload?
2. Application
Apply the concept to a case or calculation.
For [2,3,3,4,18], which pair is mean and median?
What is the mode of [A,B,B,C,C]?
3. Interpretation and limitations
Evaluate assumptions and justify conclusions.
Changing only 18 to 180 in [2,3,3,4,18] does what?
A mean of 12 minutes comes from 20 observed students. What is their total?
Explain it in writing
Self-checked, not marked. Write a few sentences, then compare with the example answer. Your writing is saved as you type.
Show an example answer
Median with IQR is often suitable because a few very large times have less effect on it. Still report the distribution and missing count.
Check: did you answer the question, use evidence or working, and avoid claiming more than the evidence supports?
Show an example answer
When total observed workload matters: observed total equals valid n times observed mean. Using a larger population n when values are missing adds an assumption.
Check: did you answer the question, use evidence or working, and avoid claiming more than the evidence supports?