Mean, Median and Mode
"What is a typical sale?" sounds like one question with one answer. It is not. There are three common ways to describe the centre of a column, and on the same data they can disagree sharply. Knowing which one to quote is the start of honest analysis.
The rule
Mean balances, median splits, mode repeats.
Back to the clean sales table, with copies 3, 5, 2, 4, 1, 6, 2, 3.
- Mean: add everything, divide by how many. The balance point.
26 / 8 = 3.25. - Median: sort and take the middle. With an even count, average the two middle values. Sorted:
1 2 2 3 3 4 5 6; the middle pair is 3 and 3, so the median is3.0. - Mode: the most frequent value. Here 2 and 3 each appear twice, so there are two modes.
sales["copies"].mean() # 3.25
sales["copies"].median() # 3.0
sales["copies"].mode() # 0 2
# 1 3mode() returns a Series, not one number, precisely because ties happen. It also works on text: sales["genre"].mode() gives fiction and science, tied on three sales each.
Now the lesson that matters. A school orders 60 copies in one bulk sale. Add it, making nine values:
with_bulk = pd.concat([sales["copies"], pd.Series([60])])
with_bulk.mean() # about 9.56
with_bulk.median() # 3.0One unusual sale nearly tripled the mean, while the median did not move. The mean feels every value's size; the median only cares about order.
Each of these works per group too: sales.groupby("branch")["copies"].median().
The shortcut
Before quoting a centre, compare mean and median. Close together: the data is roughly balanced, and the mean is fine. Far apart: something large or small is pulling the mean, so quote the median (and look at that value). Use the mode for categories, where there is nothing to add up.
Compute both
Revenue per sale is 750, 2000, 800, 1000, 250, 720, 240, 1200. The mean is 6960 / 8 = 870.
The trap is reporting the mean of skewed data, such as incomes or order sizes, as "typical". A few giants make the mean describe almost nobody.
Drills
Answer fast — trust the rule.
What is the median of 4, 1, 7, 3, 9?
What is the median of price: 250, 400, 400, 250, 250, 120, 120, 400?
Why does sales["copies"].mode() return two values?
Salaries at a small firm: seven people earn around 30,000 and the owner earns 900,000. Which figure best describes a typical salary?
Remember
- Mean balances, median splits, mode repeats.
- Compare mean and median: a big gap means something is pulling the mean.
mode()returns a Series because ties are normal.