Measures of Center
“Three rulers measure the center: mean, median, and mode.”
The formula
x̄ = (x₁ + x₂ + ⋯ + xₙ) / nHow to read it: the mean is every data value added up, then divided by how many there are
- x₁ … xₙ
- — each data value
- n
- — how many data values there are
- x̄
- — the mean (sum ÷ count)
The hook
There's more than one way to sum up a whole dataset with 'one representative value' — mean, median, mode, each the star in different situations.
In plain words
The mean is the sum divided by the count, the median is the middle value in order, and the mode is the value that shows up most.
The intuition
The mean is the balance point where the data would sit level on a seesaw. The median is whoever stands dead center once you line everyone up, and the mode is where the most values pile up. All three can point to different 'centers'.
How it's built
The mean uses every value, so extremes sway it. The median only cares about order, so it shrugs off extremes. The mode is the most common value, showing the 'trend'.
Example
Data 2, 3, 3, 7, 10 → mean = 25/5 = 5, median = (the center of 2, 3, 3, 7, 10 in order) 3, mode = the most frequent, 3.
Common mistake
The mean isn't always 'representative'. If one person's salary is huge, the mean shoots up while most people earn less — here the median is more honest.
Where it's used
Summarizing grades, incomes, heights, or view counts in a single number. For data with big extremes like income, you should look at the median too to avoid distortion.
Where it came from
The mean was used long ago to shrink errors in astronomy by 'evening out' many measurements; the median and mode were pinned down as statistics matured.
Prerequisites
Quick check
What is the median of the data 4, 6, 6, 8, 11?
- 6✓
- 7
- 8
- 35
Practice
What is the mean of the data 3, 5, 7, 9, 11?
Answer: 7
- Sum = 3+5+7+9+11 = 35
- 35 / 5 = 7
Takeaway: Mean = sum of the data ÷ count.
What is the median of the data 2, 3, 5, 8, 8, 10? (an even count)
Answer: 6.5
- In order, the two middle values are 5 and 8
- (5 + 8) / 2 = 6.5
Takeaway: With an even count, the median is the average of the two middle values.
For the data 1, 2, 2, 2, 5, 9, find the mean, median, and mode, and think about which best represents the data.
Answer: undefined
- Mean = 21/6 = 3.5
- Median = average of the two middle values (2, 2) = 2
- Mode = the most frequent, 2
- The large value 9 pulls the mean up, so the median/mode (2) is more representative
Takeaway: When extremes are present, the median can be more honest than the mean.