Which centre tells the truth?

By the end of this lesson you can choose the right centre for a dataset (mean, median, mode, weighted average, trimmed mean or geometric mean) and explain your choice in one honest sentence.

Why it matters

Every report you write will say "the average is ...". Pick the wrong centre and your report can mislead people, even though every number in it is correct. Picking the right one, and saying why, is what makes people trust an analyst.

The idea in plain words

Ask a few questions, like a detective.

First: what kind of data is it? (Lesson 4.) If it is categories with no order, like fruit or island, only the mode makes sense. If the categories have an order, like income group or small, medium and large, the median works too.

For numbers: do all the values count equally? If some values stand for more things than others, such as a price paid on 49 oranges and a price paid on 22 watermelons, use a weighted average (Lesson 7).

Are the numbers growth factors that multiply, like "up 10 per cent, then down 5 per cent"? Then use the geometric mean (Lesson 8).

Otherwise: is it lopsided, or are there a few extreme values? "Lopsided" means most values bunch up on one side and a few stretch far out on the other side, like Amara's festival day. If it is lopsided, use the median, or a trimmed mean if you want to use most of the data (Lesson 8). If it is not lopsided, or only a little, the mean and the median come out close, and the mean is fine.

A few questions pick the centre. When in doubt, work out both the mean and the median, and look at a chart. Diagram, no data.

Lopsided (statisticians say "skewed"): most values sit on one side, with a long thin tail of a few values on the other side. You will look at shapes like this in Module 3.

A quick test. Work out the mean and the median. If they are far apart compared with the size of the values, the data is probably lopsided, and the mean is usually being pulled towards the long tail. Then look at a chart to check.

See it done

Data: wdi-2024.csv (real, World Bank). Column F is life_expectancy_years: how many years a baby born in 2024 would live if that year's death rates never changed. Rows 2 to 189, one row per country or economy.

Question: what is the life expectancy of a typical country or economy?

1. Is it categories or numbers? Numbers (years).
2. Find both centres:
   • =ROUND(AVERAGE(F2:F189), 1) gives 73.9 years.
   • =MEDIAN(F2:F189) gives 74.65 years.
3. Are they far apart? No: less than one year apart, for values of around 74 years.
4. Look at the chart to be sure.

Life expectancy in 2024: a thin tail on the left makes it slightly lopsided, but only a little, so the mean and median stay close. Source: World Bank, World Development Indicators, 2024 values, CC BY 4.0. Changes: see the data credits in the module introduction.

Decide and say why: "The typical life expectancy is about 73.9 years (the mean). The median, 74.65 years, is close, so the mean is a fair summary."

Now compare GDP per person, which you met in Lesson 3. There the mean ($22,706) was about three times the median ($7,717.50), because a few very rich economies make a long tail on the right. For GDP per person, the honest choice is the median.

Same file, same countries, different columns, different right answers. The data decides, not habit.

Your turn, with help

Column G of wdi-2024.csv is basic_water_pct. It is the share of people (per cent) with basic drinking water: from an improved source (such as a tap, a borehole or a protected well), no more than 30 minutes there and back. Which centre should you report for a typical country or economy?

Hints: Find =ROUND(AVERAGE(G2:G189), 1) and =MEDIAN(G2:G189). Are they far apart? Which way is the mean pulled? Remember: no country can be above 100 per cent.

Show the answer

Mean 90.7 per cent, median 97.2 per cent. They are several points apart, and this time the mean is below the median. Many countries are close to 100 per cent, and a smaller number are much lower: the long tail is on the left, and it pulls the mean down. Report the median, and say that a group of countries is far below it.

Lopsided data can lean either way. The mean is usually pulled towards the long tail.

Your turn, alone

For each situation, choose the centre (mean, median, mode, weighted average, trimmed mean or geometric mean) and give a reason in a few words.

1. A city reports the price of a typical house. A few houses are mansions.
2. A shop's sales rose 8 per cent, fell 3 per cent, then rose 12 per cent. What was the average growth per month?
3. In penguins.csv: which island had more penguins recorded than any other?
4. Amara wants the average price customers paid per piece across all her sales, when she sells many more oranges than watermelons.

Watch out

• Using the mean on lopsided data. It is usually pulled towards the tail and can describe almost nobody (only 5 of Amara's 30 days were above her mean).
• Using the mode for measured numbers. When values rarely repeat, like flipper lengths in millimetres, the mode tells you very little.
• Saying "average" without saying which. Write "mean" or "median". Better still, give both when they differ, and say why.

Say it back

1. Which centre is the only one that works for categories with no order, like fruit names?
2. The mean is much bigger than the median. Which way is the data lopsided?
3. What quick test tells you whether data is lopsided?
4. When should you use a weighted average instead of the ordinary mean?

Show the answers

1. The mode.
2. Most likely to the right: a long tail of a few large values is pulling the mean up. Check with a chart.
3. Compare the mean and the median. If they are far apart, it is probably lopsided. Then check with a chart.
4. When some values stand for more things than others (more pieces, more pupils, more people).

What is next

You can now squeeze a pile of numbers into one honest centre, but one number is never the whole story: remember Kofi?

Amara and Kofi had exactly the same average for days 1 to 7: ₦31,914. Synthetic data.

Same average, to the naira. Yet Kofi's lowest day was ₦4,550, against Amara's ₦27,000, and he nearly ran out of money to pay his supplier. Module 2 shows you the number that sees the difference.

Formulas in this lesson were checked on 2026-10-09 (updated 2026-10-11) against the official Google Sheets and Microsoft Excel help pages, and every number was calculated by running the formulas in a spreadsheet program (LibreOffice Calc) and checked by a separate calculation. Menu steps are written for Google Sheets on the web and Excel for Microsoft 365.

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Answers to Your turn, alone

Show the answers

1. Median. A few mansions make a long tail that would pull the mean up.
2. Geometric mean. Growth rates multiply: turn them into factors (1.08, 0.97, 1.12) and use GEOMEAN.
3. Mode. Island is a category with no order. (It is Biscoe, from Lesson 6.)
4. Weighted average. Each price must count as many times as the pieces sold at that price: total money ÷ total pieces.