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Mean, Median,
and Range

Master the three essential measures of central tendency and spread that appear throughout the ISEE Quantitative Reasoning section.

ISEE Upper Level

What Are Statistical Measures?

When you have a data set, like test scores or temperatures, you need ways to summarize that data. Three of the most fundamental tools are the mean, median, and range.

Mean
The arithmetic average of all values in the set
Median
The middle value when data is ordered from least to greatest
Range
The difference between the largest and smallest values

Visualizing a Data Set

Consider the data set: 3, 7, 7, 10, 13. A dot plot shows each value on a number line, helping us see where the data clusters and how it spreads.

3 7 10 13 Mean = 8 Median = 7

Why These Measures Matter

On the ISEE, you will encounter problems that ask you to compute these measures and, even more importantly, to reason about how they change when values are added, removed, or altered.

1
Direct calculation: "Find the mean of the set \({4, 9, 11, 16}\)."
2
Reverse engineering: "The mean of 5 numbers is 12. What is the sum?"
3
Comparison: "Which is greater: the mean of Set A or the median of Set B?"

The Mean Formula

$$\text{Mean} = \frac{x_1 + x_2 + \cdots + x_n}{n}$$

Example: For the set \({5, 8, 12}\): \(\text{Sum} = 5 + 8 + 12 = 25\). There are \(n = 3\) values, so \(\text{Mean} = \frac{25}{3} \approx 8.33\).

The Median Rule

Step 1: Arrange the values in order from least to greatest.
Step 2: Find the middle value.

Odd Count
If \(n\) is odd, the median is the value at position \(\frac{n+1}{2}\).
Even Count
If \(n\) is even, the median is the average of the two middle values.

The Range

2 Min 20 Max Range = 18

Mean vs. Median: Key Distinction

The mean uses every value and is sensitive to outliers. The median only depends on position and is resistant to outliers.

Set A: {2, 4, 6, 8, 10} Median=6, Mean=6 Set B: {2, 4, 6, 8, 50} Outlier Median=6, Mean=14

How to Apply These Measures

1
Read carefully: Determine whether the problem asks for the mean, median, or range.
2
Order the data: Always sort the values from least to greatest before finding the median or range.
3
Count the values: Know \(n\). This tells you the denominator for the mean and whether the median uses one or two middle values.
4
Compute carefully: For the mean, sum all values then divide. For the median, locate the middle. For the range, subtract min from max.
5
Check your answer: Verify the mean is between the minimum and maximum. Verify the median is an actual value (or average of two values) in the ordered set.

Example 1: Direct Calculation

Find the mean, median, and range of the set \({12, 5, 8, 15, 10}\).

1
Order: \(5, 8, 10, 12, 15\)
2
Mean: \(\frac{5+8+10+12+15}{5} = \frac{50}{5} =\) \(10\)
3
Median: 5 values (odd), so the middle is the 3rd value: 10
4
Range: \(15 - 5 =\) \(10\)

Example 2: Even Number of Values

Find the median of \({3, 7, 11, 15, 20, 22}\).

3 7 11 15 20 22 Two middle values

Example 3: Reverse Engineering

The mean of 4 numbers is 15. If three of the numbers are \(10, 18,\) and \(12\), what is the fourth number?

1
Find the total sum: \(\text{Sum} = \text{Mean} \times n = 15 \times 4 = 60\)
2
Sum the known values: \(10 + 18 + 12 = 40\)
3
Subtract: \(60 - 40 =\) \(20\)

Example 4: Adding a Value

A set of 5 numbers has a mean of \(20\). If the number \(32\) is added to the set, what is the new mean?

1
Original sum: \(20 \times 5 = 100\)
2
New sum: \(100 + 32 = 132\)
3
New count: \(n = 6\)
4
New mean: \(\frac{132}{6} =\) \(22\)

Common Mistakes to Avoid

✕ Forgetting to Sort
Finding the median from an unsorted list gives the wrong middle value. Always reorder the data first.
✕ Wrong Middle for Even \(n\)
With an even count, the median is the average of the two middle values, not just one of them.
✕ Dividing by Wrong \(n\)
When a value is added or removed, the count \(n\) changes. Update the denominator before dividing.

Summary Comparison

Property Mean Median Range
What it measures Center (average) Center (middle) Spread
Formula \(\frac{\text{Sum}}{n}\) Middle value(s) Max − Min
Affected by outliers? Yes No Yes
Requires sorting? No Yes Yes
Result type Can be a decimal Value or average of two values Always a single number

Question 1 of 5

What is the mean of the set \({4, 9, 11, 16, 20}\)?

Question 2 of 5

What is the median of \({3, 7, 12, 18, 25, 30}\)?

Question 3 of 5

The mean of 6 numbers is \(10\). If one number, \(4\), is removed, what is the mean of the remaining 5 numbers?

Question 4 of 5

The set \({2, 8, x, 14, 20}\) has a mean of \(10\). What is the value of \(x\)?

Question 5 of 5

A data set has a range of \(18\) and a minimum value of \(7\). If a new value of \(30\) is added to the set, what is the new range?