# Typical value and position: mean, median, quartiles > Mean, median, mode and trimmed mean, quartiles, percentiles and the IQR (two ways to compute them), a seasonal index, and the cumulative share curve. - Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md) - Module: Averages and shares of shares - Lesson 9/36, ~27 min - Source: https://productovi.com/courses/math-basics/mean-median-trimmed-mean ## One number for a whole list **Alex Tutor:** Module 3 is about averages and shares of shares, still on Klasna's numbers. We start with the most common question in any report: what is a **typical** value? **Sofia:** Easy, the average. Ten learners took 2, 3, 4, 4, 5, 6, 7, 8, 9 and 22 lessons in April. That's 70 ÷ 10 = 7 lessons. **Alex Tutor:** Now count how many took 7 or more: only four. The learner with 22 lessons pulls the **mean** up. The mean is a balance point, so one far value moves it a lot. ![A cat watches a plank balanced on a pivot: nine small fish bunched on the left are balanced by one big orange fish at the far right end.](https://productovi.com/courses/math-basics/inline/m3-l1-i1.webp) *One heavy value far out weighs as much as many small ones.* **Alex Tutor:** See how far one value can move the mean. > **Exercise:** Ten learners took 2, 3, 4, 4, 5, 6, 7, 8, 9 and 22 lessons in April: the mean is 7. If the learner with 22 had taken 10 instead, what would the mean be? **Alex Tutor:** The **median** ignores how far the extremes are. With n = 10, even, it's the mean of the two middle values: (5 + 6) ÷ 2 = **5.5**. The **mode**, the most frequent value, is 4. **Sofia:** And mean 7 against median 5.5 tells me something? **Alex Tutor:** It's a **skew signal**: mean well above median means a long tail of large values. Revenue, lessons and time on site are usually like that. A **trimmed mean** is a compromise: drop the extremes, average the rest. Without 2 and 22: 46 ÷ 8 = **5.75**. > **Exercise:** Six learners took 3, 9, 4, 12, 6 and 8 lessons. What is the median? **Alex Tutor:** Now a trimmed mean on a different list. > **Exercise:** Seven learners took 1, 4, 5, 5, 6, 7 and 30 lessons. Drop the lowest and the highest value and average the rest. What is this trimmed mean? > **Exercise:** For learners' monthly spend, the mean is 1,850 UAH and the median is 1,240 UAH. What does that most likely mean? ## Quartiles, percentiles and the IQR **Alex Tutor:** The median splits the sorted list in two. **Quartiles** split it in four: Q₁ is the median of the lower half, Q₃ of the upper half. Lower half 2, 3, 4, 4, 5 gives Q₁ = 4; upper half 6, 7, 8, 9, 22 gives Q₃ = 8. **Alex Tutor:** The **IQR** = Q₃ − Q₁ = 4 is the width of the middle half. Unlike the full range, 22 − 2 = 20, it doesn't care about the one heavy user. ![Ten values from 2 to 22 on a line: Q₁ = 4, median 5.5 and Q₃ = 8 frame the middle half; the mean, 7, sits to the right of the median, pulled by the value 22.](https://productovi.com/courses/math-basics/diagrams/m3-l1-box.png) *The box holds the middle half; the mean is pulled toward the tail.* **Sofia:** My spreadsheet says Q₃ = 7.75, not 8. **Alex Tutor:** It interpolates. QUARTILE.INC puts Q₃ at position 1 + (n − 1) × 0.75 = 7.75 in the sorted list: three quarters of the way from x₍₇₎ = 7 to x₍₈₎ = 8. Both methods are accepted; just say which one you used. **Alex Tutor:** A **percentile** generalizes this: the 90th percentile is the value below which 90% of the list falls. Quartiles are the 25th and 75th percentiles, the median the 50th. > **Exercise:** Eight tutors gave 1, 3, 4, 6, 7, 9, 10 and 14 trial lessons. Using the halves method, what is the IQR? **Alex Tutor:** And the spreadsheet's version of a quartile. > **Exercise:** A spreadsheet's QUARTILE.INC puts Q₃ at position 1 + (n − 1) × 0.75 in the sorted list and interpolates between neighbors. For 1, 3, 4, 6, 7, 9, 10, 14, what Q₃ does it give? ## Seasons and cumulative shares **Alex Tutor:** Averages also tame seasons. A **seasonal index** is a month divided by the average month. Klasna averages 6,000 lessons a month over the year; May brings 9,620, so its index is 9,620 ÷ 6,000 ≈ **1.60**: 60% above a typical month. **Sofia:** So a May with 9,620 isn't growth, it's just May. **Alex Tutor:** Exactly. Last tool: the **cumulative share curve**, the share of learners with x lessons or fewer. For our ten: 40% took 4 or fewer, 60% took 6 or fewer, 90% took 9 or fewer. The curve crosses 50% at the median. **Alex Tutor:** Put a second group on the same chart, and where the curves cross tells you something. Below the crossing, the higher curve has more light users; above it, the other group catches up. ![Two step curves of the share of learners with x lessons or fewer: group A is higher up to 7 lessons, the curves meet at 70% at 7 lessons, and group B is higher after that.](https://productovi.com/courses/math-basics/diagrams/m3-l1-cumulative.png) *Read the curve as "this share took x or fewer".* > **Exercise:** Klasna averages 6,000 lessons a month. August brings 3,900. What is August's seasonal index? Give it to two decimals. > **Exercise:** On a cumulative share curve ("share with x or fewer"), the value of x where the curve reaches 50% is the median. ## Your turn: which number goes in the report **Alex Tutor:** Leo wants one number for "lessons per learner in April" on the investor slide. You have mean 7, median 5.5, trimmed mean 5.75 and Q₁–Q₃ of 4–8. **Sofia:** The 22 makes me nervous about the mean. > **Exercise:** For "lessons per learner in April" you have the list 2, 3, 4, 4, 5, 6, 7, 8, 9, 22: mean 7, median 5.5, trimmed mean 5.75, Q₁ = 4, Q₃ = 8. Which number would you put on the slide, and why? What one extra detail would you add so the number isn't misread? ## Where you will meet this > **In short** > > - **Mean** = sum ÷ n, a balance point that far values drag. **Median**: the middle of the sorted list (mean of the two middles for even n). **Mode**: the most frequent value. > - Mean well above median signals a long right tail. A **trimmed mean** drops the extremes first. > - **Q₁** and **Q₃** are the medians of the lower and upper halves; **IQR** = Q₃ − Q₁. Spreadsheets interpolate (QUARTILE.INC), so results can differ slightly: say which method. > - The **p-th percentile** has p% of the list below it. > - **Seasonal index** = month ÷ average month. A **cumulative share curve** shows the share with x or fewer; it crosses 50% at the median. > **These lessons use it** > > - [Mean, median and mode](/courses/statistics-for-pm-and-marketers/typical-value): which typical value to report. > - [Measures of spread](/courses/statistics-for-pm-and-marketers/spread): quartiles, percentiles and the IQR, by halves and in a spreadsheet. > - [Outliers and skewed data](/courses/statistics-for-pm-and-marketers/outliers-and-skew): the mean–median gap as a skew signal. > - [DAU, WAU, MAU and stickiness](/courses/product-analytics/active-users): averaging daily counts over a month. > - [Traffic mix and seasonality](/courses/product-analytics/traffic-and-seasons): comparing months that aren't alike. > - [A year, not a month](/courses/personal-finance/a-year-not-a-month): strong and weak months against the average month. **Alex Tutor:** Next time: the weighted mean, where some values count more than others. - Previous lesson: [Inequalities, intervals and formulas with cases](https://productovi.com/courses/math-basics/inequalities-and-intervals.md) - Next lesson: [Weighted mean and mix](https://productovi.com/courses/math-basics/weighted-mean-and-mix.md) --- Productovi Learning. 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