👀 This is a real lesson, shown the way a grade HS (12) student sees it before practicing.

🧮

Math · grade HS (12) · lesson

Describing data: center and spread (Statistics)

A few numbers can sum up a whole data set: one for its center and one for its spread. Picking the right ones depends on the data's shape.

Card 1 of 6

The mean

💡 The mean is the total divided by the count, so the total is the mean times the count.

To find the mean, add all the values and divide by how many there are.

Turned around, mean × count = total. That lets you find a missing value: find the total you need, then subtract the values you have.

🧠 Let's work it out

Four test scores have a mean of 70. Three are 65, 72, and 68. Find the fourth.

Total needed = 70 × 4 = 280.
Total so far = 65 + 72 + 68 = 205.
280 − 205 = 75.

✏️ What your answer looks like: Type just 75.

What do you think comes next?

mean: the average: total ÷ count

🔗 Khan Academy: Summarizing quantitative data · Khan Academy (free) ↗

Median and quartiles

💡 Sort first; the median splits the data in half, and the quartiles are the medians of each half.

The median is the middle value after sorting. With an even count, average the two middle values.

Q1 is the median of the lower half. Q3 is the median of the upper half. With an odd count, leave the median itself out of both halves.

The interquartile range is IQR = Q3 − Q1. It is the spread of the middle half of the data.

🧠 Let's work it out

Find the IQR of 2, 6, 3, 9, 7, 11, 5, 8.

Sort: 2, 3, 5, 6, 7, 8, 9, 11.
Lower half 2, 3, 5, 6 gives Q1 = 4.
Upper half 7, 8, 9, 11 gives Q3 = 8.5.
IQR = 8.5 − 4 = 4.5.

✏️ What your answer looks like: Type 4.5.

What do you think comes next?

median: the middle value of sorted data
quartile: a value that marks off a quarter of the sorted data
interquartile range: IQR = Q3 − Q1, the spread of the middle half

Range and outlier fences

💡 The range is largest minus smallest, and values beyond Q3 + 1.5 × IQR or Q1 − 1.5 × IQR count as outliers.

The range is the simplest spread: the largest value minus the smallest.

An outlier is a value far from the rest. The 1.5 × IQR rule draws fences. The upper fence is Q3 + 1.5 × IQR. The lower fence is Q1 − 1.5 × IQR.

🧠 Let's work it out

A data set has Q1 = 12 and Q3 = 18. Find the outlier fences.

IQR = 18 − 12 = 6.
1.5 × 6 = 9.
Upper fence = 18 + 9 = 27.
Lower fence = 12 − 9 = 3.

✏️ What your answer looks like: If asked for the upper fence, type 27.

What do you think comes next?

range: largest value minus smallest value
outlier: a value far away from the rest of the data

Shape decides the best center

💡 Extreme values pull the mean toward them, so use the median for skewed data or data with outliers.

The mean uses every value, so one huge value drags it up. The median only cares about the middle, so it barely moves.

A skewed-right graph has a long tail toward big values. The tail pulls the mean above the median. Skewed left does the opposite. In symmetric data the mean and median are close.

🧠 Let's work it out

Salaries at a small company: most near $50,000, and the owner's is $900,000. Which better describes a typical salary, the mean or the median?

The owner's salary is an extreme value.
It pulls the mean far up, but the median stays near $50,000.
So the median better describes a typical salary.

✏️ What your answer looks like: Pick that choice.

What do you think comes next?

skewed: lopsided, with a long tail on one side
symmetric: the same shape on both sides of the center

Standard deviation

💡 Standard deviation is about how far a typical value sits from the mean.

Find the mean. Find each value's distance from the mean and square it. Add the squares.

For a population, divide by n, the count. For a sample, divide by n − 1. Then take the square root.

🧠 Let's work it out

Find the population standard deviation of 1, 3, 5, 7.

Mean = 16 ÷ 4 = 4.
Squared distances from 4: 9, 1, 1, 9.
Add them: 9 + 1 + 1 + 9 = 20.
Divide by 4: 20 ÷ 4 = 5.
Take the root: √5 ≈ 2.24.
Another example: as a sample, divide by 3 instead: √(20 ÷ 3) ≈ 2.58.

✏️ What your answer looks like: Round as the question asks, like 2.2 to the nearest tenth.

What do you think comes next?

standard deviation: a measure of how spread out values are from the mean
population: every member of the group you care about
sample: a part of the group, used to learn about the whole

Shifting every value

💡 Adding the same number to every value moves the center but leaves the spread unchanged.

If everyone's score goes up 10, the mean and median go up 10. The values all move together, so the distances between them don't change.

That means the range, IQR, and standard deviation stay the same. Multiplying every value by a number is different: it scales both the center and the spread.

🧠 Let's work it out

Every value in a data set is cut by 3. What changes?

The mean and median each drop by 3.
The gaps between values don't change.
So the range, IQR, and standard deviation stay the same.

✏️ What your answer looks like: Choose the statements that match.

What do you think comes next?

center: a typical value, like the mean or median
spread: how far apart the values are, like the range or standard deviation