# Mean, median and mode: the typical value

> Mean, median, mode and trimmed mean on real-looking CatChow orders: what each one answers, why one bulk order can fool the average, and how not to average averages.

- Course: [Statistics for Product Managers and Marketers](https://productovi.com/courses/statistics-for-pm-and-marketers.md)
- Module: Describing data
- Lesson 1/24, ~48 min
- Source: https://productovi.com/courses/statistics-for-pm-and-marketers/typical-value

## Where you work

**CatChow** is a small online shop that sells cat food, treats and litter and delivers them to the door. Its team of about thirty has hit a familiar problem: dashboards full of averages, rates and charts that different people read in different ways. You've just joined the product team. In this course you'll learn to read those numbers honestly: describe the data, compare groups, run A/B tests you can trust, look for relationships and segments, and say how sure you are. This first lesson starts with the most common question in any meeting, what counts as a typical value, and answers it with the mean, median and mode.

![A newcomer with a laptop is welcomed by three smiling teammates at a table with cat-food boxes, while an orange office cat watches from the table.](https://productovi.com/courses/statistics-for-pm-and-marketers/inline/typical-value-illustration-1.webp)

*Your first day on CatChow's product team.*

## "So what's our typical order?"

It's Monday stand-up at **CatChow**. The marketing lead is planning a free-shipping offer and asks one simple question: *what does a typical order look like?*

You open last week's orders: 20 of them, from \$18 to \$284. Nobody wants to hear twenty numbers, so you need **one number that stands for all of them**. Statisticians call such a number a **measure of central tendency**. There are several, and they can give very different answers from the same data. Choosing the right one is a product decision, not a formality.

In this lesson you'll learn four of them (the **mean**, the **median**, the **mode** and the **trimmed mean**), plus a trap that catches almost every dashboard: averaging averages. You can download the week of orders below and follow along in Google Sheets or Excel.

**Dataset (CSV):** [CatChow orders, 14–20 September](https://productovi.com/datasets/statistics-for-pm-and-marketers/typical-value/catchow-orders-week.csv)

20 orders placed in one week. Synthetic data made for this course.

- `order_id`: Order number
- `order_date`: Date the order was placed (YYYY-MM-DD)
- `order_value`: Order value, USD

## The mean: share the total equally

The **mean** (arithmetic mean) is what most people call "the average": add up all the values and divide by how many there are. Last week CatChow took \$900 across 20 orders, so the mean order value is \$900 / 20 = \$45.

In e-commerce this number has its own name: **average order value (AOV)**. It is the mean by definition, total revenue divided by the number of orders.

The mean has one great strength. It is tied to the total: mean × count = total, always. If you expect 400 orders next month and the mean stays at \$45, you can forecast \$18,000 of revenue. No other measure in this lesson lets you do that.

$$
\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n} = \frac{1}{n}\sum_{i=1}^{n} x_i
$$

*The mean: the sum of all n values divided by n. The bar over x reads "x-bar".*

![A cat with a scoop shares one heap of kibble evenly across a row of identical blue bowls, all filled to the same level; a large orange sack feeds the heap.](https://productovi.com/courses/statistics-for-pm-and-marketers/inline/typical-value-illustration-2.webp)

*The mean asks how big each share would be if the total were split equally.*

Now look at the orders themselves. **16 of the 20 orders are below \$45.** How can the "typical" order be bigger than almost every real order?

Scroll the file and you'll find order CC-1047: **\$284**. A local cat shelter stocked up for the month. That single order adds \$284 to the total, and because the mean shares the total equally, it raises every order's "fair share" by about \$12.6. The mean answers "how much would each order be if they were all equal?", and that is not the same question as "what does a normal order look like?"

## How to find the median: the middle of the line

The **median** ignores the sizes of the extremes and only asks about order. Sort the values from smallest to largest and take the one in the middle. Half of the orders are smaller, half are larger.

With an odd number of values there is exactly one middle value: for 7 values it is the 4th. With an even number there are two, so you average them. Our 20 sorted orders have 31 in 10th place and 33 in 11th place, so the median is $(31 + 33) / 2 = 32$ dollars.

In Sheets or Excel: `=MEDIAN(C2:C21)`. By hand, the only real work is sorting.

![Cats stand in a line sorted from the smallest kitten to the largest cat; the one in the exact middle of the line is highlighted in orange.](https://productovi.com/courses/statistics-for-pm-and-marketers/inline/typical-value-illustration-3.webp)

*Sort first, then take the one in the middle.*

$$
\operatorname{Me} = \begin{cases} x_{(k+1)}, & n = 2k + 1 \\[6pt] \dfrac{x_{(k)} + x_{(k+1)}}{2}, & n = 2k \end{cases}
$$

*Values are sorted from smallest to largest; x₍ₖ₎ is the k-th of them. Top line: odd n; bottom line: even n.*

**Last week's orders by value**

| Order value, USD | Orders |
| --- | --- |
| 10–19 | 1 |
| 20–29 | 7 |
| 30–39 | 7 |
| 40–49 | 4 |
| 50–279 | 0 |
| 280–289 | 1 |

*Most orders sit between 20 USD and 40 USD. One bulk order of 284 USD sits far to the right and drags the mean up to 45 USD, while the median stays at 32 USD.*

## One order, two very different reactions

Here is the clearest way to see the difference. Remove the shelter's order and recompute:

| | All 20 orders | Without CC-1047 |
|---|---|---|
| Mean | \$45.00 | \$32.42 |
| Median | \$32 | \$31 |

One order out of twenty moved the mean by more than \$12 but moved the median by \$1. We say the median is **robust**: a few extreme values barely affect it. The mean is **sensitive**: it reacts to every single value, including the weird ones.

Neither behaviour is "better". If the question is *how much revenue did we make?*, the shelter's \$284 is real money and belongs in the answer, so you want the mean. If the question is *what does a normal customer put in the basket?*, the shelter is not a normal customer, so you want the median.

> **Report both, watch the gap**
>
> When mean and median are close, the data is roughly balanced and either will do. When the mean is far above the median, a few large values are pulling it: look at them before you trust any average. The gap itself is a cheap early-warning signal on any dashboard.

## What is the mode in statistics?

The **mode** is the most common value: the one that appears most often. Among last week's orders, \$28 appears three times, more than any other value, so the mode is \$28.

For money and time, the mode is usually the least useful measure. With precise values (like \$31.47) almost nothing repeats, and a tiny change in the data can move the mode somewhere else entirely. Here \$24 and \$48 each appear twice, only one order behind.

The mode shines with **categories**, where mean and median make no sense at all. What is the most popular flavour, salmon or chicken? Which plan do most subscribers choose? What rating do users give most often? You can't average "salmon" and "chicken", but you can count them. The mode is the only measure of central tendency that works for categories.

Watch out for data with **two modes**. If a histogram of basket sizes has one peak around \$25 and another around \$120, you probably have two different kinds of customers (households and, say, shelters or cat cafés). Any single "typical value" would describe neither group. That's a hint to segment, not to average.

## The trimmed mean: a compromise

Sometimes you want something that behaves like a mean but doesn't let a handful of extremes run the show. The **trimmed mean** does exactly that: sort the data, drop a fixed share from *each* end (often 5–10%), and take the mean of what's left.

With a 10% trim on 20 orders, you drop 2 from the bottom (\$18, \$22) and 2 from the top (\$48, \$284). The remaining 16 orders add up to \$528, so the trimmed mean is \$528 / 16 = \$33. That's close to the median, but it still uses the actual values of 16 orders instead of just the middle two.

You have met this idea before: in some judged sports the highest and lowest scores are thrown away before averaging, so that one overly generous or harsh judge can't swing the result. In Sheets or Excel: `=TRIMMEAN(C2:C21, 0.2)`. Note that the function takes the *total* share to remove, so 0.2 means 10% from each end.

$$
\bar{x}_{[k]} = \frac{1}{n - 2k}\sum_{i=k+1}^{n-k} x_{(i)}
$$

*Trimmed mean: sort the values, drop k from each end, average the remaining n − 2k.*

## The trap: averaging averages

One more mean, because it causes real mistakes in product reviews. Suppose CatChow's checkout conversion last month was **2%** on mobile (8,000 sessions) and **5%** on desktop (2,000 sessions). What is the overall conversion rate?

The tempting answer is $(2\% + 5\%) / 2 = 3.5\%$. It's wrong. Mobile had four times the traffic, so it should count four times as much. Count the orders instead: $8{,}000 \times 0.02 = 160$ on mobile plus $2{,}000 \times 0.05 = 100$ on desktop gives 260 orders from 10,000 sessions, which is **2.6%**.

This is a **weighted mean**: each group's value counts in proportion to its size. Whenever you combine rates or averages from segments (by channel, country, device, cohort), weight them by the segment's size, or simply go back to the raw totals and divide.

![A balance scale holds a big pile of smartphones on one pan and a few laptops on the other; it tips toward the phones while a small cat looks on.](https://productovi.com/courses/statistics-for-pm-and-marketers/inline/typical-value-illustration-4.webp)

*Bigger groups must weigh more when you combine rates.*

$$
\bar{x}_w = \frac{\sum_{i} w_i x_i}{\sum_{i} w_i}
$$

*Weighted mean: multiply each value by its weight (e.g. sessions), add up, divide by the total weight.*

## When to use the mean, median or mode

| Your question | Use | Why |
|---|---|---|
| How much revenue will N orders bring? | Mean | Only the mean links to totals |
| What does a typical order or user look like? | Median | Robust to a few extreme values |
| Which category is most popular? | Mode | The only option for categories |
| A mean-like number that resists a few freak values | Trimmed mean | Keeps most of the data, drops the extremes |
| Combining rates across segments of different size | Weighted mean | Big segments must count more |

A habit worth building: whenever someone shows you "the average", ask **which average, and computed over what?** Most tools default to the mean, and most product data (order values, session lengths, revenue per user) has a long right tail. In the next lessons you'll learn to measure how spread out the data is, and how to spot and handle the extreme values that made our mean misbehave.

## Exercises

Answer them on the lesson page to get feedback.

1. CatChow's product pages usually load in 1–2 seconds, but about 3% of loads hit a 30-second timeout. You want one number that describes **what a typical visitor experiences**. Which do you report?

2. If the mean order value is higher than the median, then most orders are above the mean.

3. Match each question to the measure that answers it best.

4. Download the dataset **CatChow orders, 14–20 September**. What is the median order value?

5. In the dataset **CatChow orders, 14–20 September**, remove the shelter's bulk order CC-1047 (\$284). What is the mean of the remaining 19 orders? Round to cents.

6. Which statements about the **mean** are true? Select all that apply.

7. Last month CatChow's newsletter traffic converted at **4%** over 1,500 sessions, and paid-social traffic at **1%** over 4,500 sessions. What is the overall conversion rate across both channels?

8. Last week CatChow's mean order was \$45 and the median was \$32. Which sentence belongs in the weekly report?

9. A partner pet shop sent CatChow 10 orders last week: \$12, \$20, \$22, \$25, \$27, \$30, \$31, \$35, \$40 and \$190. What is the **10% trimmed mean** of these orders?

10. A weekly report says: *"Average order value jumped from \$38 to \$52 this week!"* Before celebrating, what would you check, and why? Answer in 2–4 sentences.

11. What free-shipping threshold would you recommend, and why? Use numbers from the dataset.

- Next lesson: [Measures of spread: range, SD and quartiles](https://productovi.com/courses/statistics-for-pm-and-marketers/spread.md)

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