# Outliers and skewed data: how to find them

> Why product metrics have long tails, how to find outliers with the IQR rule, and what to do with them: fix errors, segment different customers, and never delete whales without thinking.

- Course: [Statistics for Product Managers and Marketers](https://productovi.com/courses/statistics-for-pm-and-marketers.md)
- Module: Describing data
- Lesson 3/24, ~44 min
- Source: https://productovi.com/courses/statistics-for-pm-and-marketers/outliers-and-skew

## The customer who broke the average

Another week on CatChow's product team, another number to check. The growth team is preparing a quarterly review. One slide shows the **90-day spend of new customers**: how much each customer who joined in June spent in their first 90 days. The analyst took a random sample of 24 of them (download it below).

The mean is **\$50**. The median is **\$35**. Two-thirds of the customers (16 of 24) spent less than the mean. And one customer, U-5120, spent **\$240**, almost five times the mean.

Someone suggests: *"Just delete the top values, they're outliers."* Should you? This lesson is about **outliers** (values far from the rest) and how to find them, and about **skewed distributions** (data with a long tail on one side). They are the norm in product and marketing data, not the exception, and handling them well is one of the most practical skills in this course.

**Dataset (CSV):** [90-day spend of new CatChow customers](https://productovi.com/datasets/statistics-for-pm-and-marketers/outliers-and-skew/catchow-customer-spend.csv)

A random sample of 24 customers who joined in June: spend and number of orders in their first 90 days. Synthetic data made for this course.

- `customer_id`: Customer ID
- `spend_90d`: Total spend in the first 90 days, USD
- `orders_90d`: Number of orders in the first 90 days

**90-day spend per new customer**

| Spend in 90 days, USD | Customers |
| --- | --- |
| 0–19 | 3 |
| 20–39 | 11 |
| 40–59 | 5 |
| 60–79 | 1 |
| 80–99 | 2 |
| 100–119 | 1 |
| 120–239 | 0 |
| 240–259 | 1 |

*A long right tail: most customers spent 20–60 USD, a few spent much more, and one spent 240 USD. The tail pulls the mean (50 USD) to the right of the median (35 USD).*

## What is an outlier? Three different kinds

An **outlier** is an observation that sits far away from the rest of the data. That's a statistical description, not a verdict. Before deciding what to do, ask *why* the value is so far out. In practice there are three very different reasons:

1. **It's an error.** A test order placed by a developer, a price typed as \$2,990 instead of \$29.90, revenue recorded in the wrong currency, a bot that added 400 items to a basket. These values describe nothing real. Fix or remove them, and write down that you did.
2. **It's a genuine extreme.** A real customer who really loves your product and orders a lot. The value is correct; it's just rare. Deleting it means pretending your best customers don't exist.
3. **It's from a different population.** Look at U-5120 again: \$240 over **12 orders** in 90 days, while almost everyone else placed 1–3 orders. That pattern suggests a business (a cat café, a shelter, a small reseller) rather than a household. It's real data, but it answers a different question. Analyse it as a separate segment rather than mixing it with households.

Outliers are information. Sometimes the most important finding in a dataset is the outlier itself: a B2B segment you didn't know you had, or a tracking bug that has been inflating revenue for a month.

![Three cats stand apart from a crowd: one tangled in a sparking cable, one hugging a huge bag of food, one in an apron beside a row of shelter bowls.](https://productovi.com/courses/statistics-for-pm-and-marketers/inline/outliers-and-skew-illustration-2.webp)

*An error, a genuine extreme, or a different kind of customer.*

## How to find outliers: the IQR rule

"Far from the rest" needs a definition, or everyone will draw the line wherever suits their story. The most common rule, proposed by the statistician John Tukey, builds **fences** from the quartiles you met in the previous lesson:

- lower fence: $Q_1 - 1.5 \times \text{IQR}$
- upper fence: $Q_3 + 1.5 \times \text{IQR}$

Anything outside the fences is flagged as a potential outlier. For our 24 customers, $Q_1 = 24$ and $Q_3 = 56$, so $\text{IQR} = 32$. The upper fence is $56 + 1.5 \times 32 = 104$ dollars, and the lower fence is $24 - 48 = -24$ dollars (spend can't be negative, so nothing is flagged at the bottom).

Two customers are above \$104: **U-5115 (\$110)** and **U-5120 (\$240)**. Note that U-5115 spent \$110 over 4 orders, which looks like an enthusiastic household rather than a business. The rule flags values for a closer look; it doesn't tell you what they are.

Because the fences are built from quartiles, the outliers themselves can't distort them. That is the rule's biggest strength. The same fences decide which points a box plot draws as separate dots, as you'll see in the next lesson.

$$
x \notin \left[\, Q_1 - \tfrac{3}{2}\,\text{IQR},\;\; Q_3 + \tfrac{3}{2}\,\text{IQR} \,\right]
$$

*Tukey's rule: a value outside the fences (3/2 = 1.5) is a potential outlier. Values beyond 3 × IQR are sometimes called extreme outliers.*

![Many cats sit inside a light blue fenced yard; two sit outside the fence, one far away and lit by an orange spotlight.](https://productovi.com/courses/statistics-for-pm-and-marketers/inline/outliers-and-skew-illustration-3.webp)

*Outside the fence means take a closer look, not delete.*

## Why "3 standard deviations" can fail you

Another popular rule flags values more than 3 SDs from the mean. It works reasonably well for large, bell-shaped datasets. On small or skewed data it has a nasty flaw: **the outliers inflate the very SD you measure them with.**

Our sample has a mean of \$50 and an SD of about \$47.7. U-5120 is $(240 - 50) / 47.7 \approx 3.99$ SDs above the mean, so it gets flagged, but only just, and mostly because it is so extreme. U-5115 is only 1.26 SDs above the mean and looks perfectly ordinary.

Now remove U-5120 and recompute: the mean falls to \$41.74 and the SD to about \$25.7. Suddenly U-5115 is 2.65 SDs out. One extreme value was hiding another. Statisticians call this **masking**. The IQR rule doesn't suffer from it, which is why it is the better default for everyday product data.

![A giant orange cat stands in front of a smaller blue cat and hides it from a magnifying glass; only the hidden cat's tail pokes out.](https://productovi.com/courses/statistics-for-pm-and-marketers/inline/outliers-and-skew-illustration-4.webp)

*A huge value can hide a smaller one standing behind it.*

## Right and left skew: when the tail wags the mean

Look at the chart again. The values pile up on the left and trail off in a long tail to the right. That shape is called **right-skewed** (or positively skewed). In a skewed distribution, the few values in the tail pull the mean towards them, while the median stays with the crowd:

- **Right skew** (long tail to the right): mean > median. Here \$50 > \$35.
- **Symmetric**: mean ≈ median.
- **Left skew** (long tail to the left): mean < median.

Right skew is everywhere in product and marketing data, because these metrics have a hard floor at zero but no ceiling: revenue per customer, order value, session length, time to first purchase, number of orders, followers per account, ad cost per acquisition. If a metric is "how much" or "how long" and can't go negative, expect a right tail.

Left skew is rarer, but you'll meet it in app-store ratings (most people give 5 stars, with a tail of angry 1s) and in "percent of lesson completed" when most people finish.

The mean-versus-median check is a rule of thumb, not a law; odd shapes can break it. When in doubt, plot the histogram.

> **Deleting whales can delete your business**
>
> In our sample the top 2 customers (8% of 24) brought \$350 of \$1,200, or **29% of all revenue**. Heavy-tailed data means a small group of customers often drives a large share of the money. If you delete them to "clean" the data, your revenue forecast will be badly wrong, and you'll miss the segment most worth protecting.

## What to do with outliers: a checklist

1. **Look before you touch.** Open the raw rows. Check dates, IDs, order counts, currency. Is this a bug, a real person, or a different kind of customer?
2. **Fix or remove errors, and log it.** Test orders, duplicates and typos go. Write down what you removed and why, so that someone else gets the same numbers.
3. **Choose the summary by the question, not by convenience.**
   - *"How much revenue will the June cohort bring?"* Totals need the **mean**, with the whales included.
   - *"What does a typical new customer spend?"* Use the **median** (here \$35) or a **trimmed mean**.
   - *"Is our spend distribution changing?"* Track the median and a high percentile (p90) together.
4. **Segment instead of deleting.** Pull likely businesses (many orders, large baskets) into their own segment. Households and cafés behave differently and deserve separate metrics.
5. **Use robust tools when extremes aren't the point.** Median, IQR, trimmed mean, or *capping*: replace values above, say, the 99th percentile with the 99th percentile itself (also called winsorising). Capping is common in A/B tests on revenue, where one huge order can otherwise decide the result.
6. **Report with and without.** "Mean 90-day spend is \$50, or \$41.74 excluding one business customer." Two numbers, full honesty, and no one can accuse you of cherry-picking.

> **A quick skew check for any dashboard**
>
> Divide the mean by the median. Around 1: roughly symmetric. Well above 1 (here \$50 / \$35 ≈ 1.4): a right tail that deserves a look. It takes ten seconds and catches most "the average jumped!" false alarms.

## Exercises

Answer them on the lesson page to get feedback.

1. Which CatChow metric is most likely to be **right-skewed**?

2. Every outlier should be removed before you analyse the data.

3. Match each unusual value to the most sensible action.

4. A dataset of monthly ad clicks per campaign has $Q_1 = 120$ and $Q_3 = 200$. Using Tukey's rule, what is the **upper fence** above which a campaign is flagged as a potential outlier?

5. Using the dataset **90-day spend of new CatChow customers**, what share of total 90-day spend (in %) comes from the two customers flagged by the IQR rule? Round to one decimal.

6. Time to first purchase after sign-up: mean 19 days, median 6 days. What shape does this distribution most likely have?

7. In the 90-day spend sample, the mean is \$50 with all 24 customers and \$41.74 without U-5120, a likely business. Which line belongs on the quarterly review slide?

8. Instead of deleting the two flagged customers, the analyst **caps** 90-day spend at the upper fence, \$104: U-5115 (\$110) and U-5120 (\$240) are both set to \$104, and nobody is removed. What is the new mean of the 24 customers? Round to cents.

9. You have 30 values of revenue per customer with a long right tail. Which rule is the safest default for flagging potential outliers?

10. Explain in 2–4 sentences why the "more than 3 SDs from the mean" rule can miss outliers in a small, skewed dataset.

11. Which number should go into the acquisition budget, and how do you handle the outliers? Use numbers from the dataset.

- Previous lesson: [Measures of spread: range, SD and quartiles](https://productovi.com/courses/statistics-for-pm-and-marketers/spread.md)
- Next lesson: [Types of charts: choosing the right one](https://productovi.com/courses/statistics-for-pm-and-marketers/charts.md)

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