Averages & Statistical Measures
A complete, test-focused breakdown of every Averages concept on the ACT — mean, median, mode, range, weighted averages, missing value problems, combined averages, and average rate of change — with worked examples and strategy notes for every question type.
In This Lesson
The Four Measures at a Glance
The ACT tests four core statistical measures. Know each one cold — the definitions are simple but the ACT disguises them in multi-step problems designed to catch careless errors.
Quick memory trick: Mean = Math (you calculate it). Median = Middle (you find it by position). Mode = Most (you count frequency). Range = Reach (how far the data stretches).
Arithmetic Mean (Average)
The mean is the standard "average" everyone learns in school. It is the single most important concept in this lesson — most ACT average problems ultimately reduce to the mean formula used cleverly.
Sum = Mean × Count
Count = Sum / Mean
// Know all three rearrangements — the ACT uses all of them
On every ACT average problem, convert the average back to a sum as early as possible. Most problems become straightforward once you work with sums rather than averages directly.
Properties of the Mean
- The mean is pulled toward outliers (extremely high or low values).
- If you add a constant k to every value, the mean increases by k.
- If you multiply every value by k, the mean is multiplied by k.
- If you add a value equal to the current mean, the mean does not change.
- The sum of the deviations from the mean always equals zero: Σ(xᵢ − mean) = 0.
ACT trap — the "average of averages" error: If Group A has 3 people averaging 70 and Group B has 7 people averaging 90, the combined average is NOT (70+90)÷2 = 80. You must weight by group size. See Section 08 for the correct method.
Median
The median is the middle value of a dataset when arranged in order from least to greatest. It is resistant to outliers — extreme values do not pull the median the way they pull the mean.
Odd number of values: Median = the single middle value.
Position = (n + 1) / 2
Even number of values: Median = mean of the two middle values.
Average positions n/2 and (n/2 + 1)
Odd Count — Single Middle Value
Dataset: 3, 7, 9, 12, 18, 22, 25 (7 values — odd)
Position: (7+1)/2 = 4th value → Median = 12
Even Count — Average the Two Middle Values
Dataset: 5, 11, 14, 18, 23, 30 (6 values — even)
Two middle values (3rd and 4th): Median = (14 + 18)/2 = 16
Must sort first! This is the most common median mistake — finding the middle value of unsorted data. On the ACT, data is often presented in a table or stem-and-leaf plot not in order. Always sort before finding the median position.
When to Use Median vs. Mean
| Situation | Best Measure | Why |
|---|---|---|
| Symmetric, no outliers | Mean | Uses all data; most informative |
| Skewed data or outliers | Median | Not affected by extreme values |
| Income / home prices | Median | A few millionaires distort the mean |
| Test scores (no outliers) | Mean | Captures overall performance level |
Mode
The mode is the value (or values) that appear most frequently in a dataset. It is the least computationally demanding measure — you simply count frequencies.
{3, 5, 5, 7, 8} → Mode = 5
{2, 4, 4, 6, 6, 9} → Mode = 4 and 6
{1, 3, 5, 7} → No mode
No sorting or calculation required.
ACT context: Mode questions are usually straightforward, but the ACT sometimes hides them in frequency tables, histograms, or dot plots. The strategy is the same: find the value with the tallest bar or highest listed frequency.
Range
The range measures the spread of a dataset. It is the simplest measure of variability and requires only two values from the entire dataset.
Range vs. Mean/Median
Range tells you how spread out the data is, not where the center is. Two datasets can have the same mean but very different ranges — the ACT sometimes exploits this distinction.
Range limitation: Because range uses only the two extreme values, a single outlier dramatically changes it. The ACT may ask you to identify which measure is most affected when an outlier is added — the answer is usually the mean and range, while the median is most resistant.
Missing Value Problems
Missing value problems are the most common averages question type on the ACT. You are given the average and all but one value, and asked to find the missing one. The key is always the same: work through the sum.
Step 1: Find the required total sum: Sum = Mean × Count
Step 2: Find the sum of the known values.
Step 3: Missing value = Required sum − Known sum
Missing Median Problems
The ACT also asks for a missing value that will produce a given median. The approach: place the unknown in the sorted list and determine what value would land in the median position.
For missing median problems: First establish the sorted order including the unknown, then identify which position is the median position, then set that position equal to the target median and solve. Always verify by re-inserting your answer into the sorted list.
Weighted Averages
A weighted average accounts for the fact that some values count more than others. Each value is multiplied by its weight (its relative importance), then all weighted values are summed and divided by the total weight.
// Σ means "sum of all"
| Component | Score | Weight | Score × Weight |
|---|---|---|---|
| Homework | 90 | 0.20 | 90 × 0.20 = 18.0 |
| Midterm | 74 | 0.30 | 74 × 0.30 = 22.2 |
| Final Exam | 82 | 0.50 | 82 × 0.50 = 41.0 |
| Total | — | 1.00 | 81.2 |
Weights must sum to 1 (or 100%). If given weights in percentages (20%, 30%, 50%), convert to decimals first. If weights are counts (40, 25, 15 items), the denominator is the total count (80). Either way, the formula is the same: Σ(value × weight) ÷ Σ(weights).
Combined Group Averages
When two or more groups are merged and you want the overall average, you cannot simply average the group averages — you must account for group sizes. This is one of the most frequently tested ACT traps.
= (Mean₁ × Count₁ + Mean₂ × Count₂) / (Count₁ + Count₂)
The "average of averages" (75+85)÷2 = 80 would only be correct if both groups were the same size. Since Class B is larger (30 vs 20), it pulls the combined average toward 85. The correct answer of 81 is closer to Class B's mean — exactly as expected.
Effect of Adding or Removing Values
The ACT frequently asks how the mean, median, mode, or range change when a value is added, removed, or altered. Think through each measure separately using the sum-and-count framework.
Adding a Value
- If the new value equals the current mean → the mean stays the same; count increases by 1.
- If the new value is above the mean → the mean increases.
- If the new value is below the mean → the mean decreases.
- The median may or may not change — must re-sort to check.
- The range may increase if the new value is above the max or below the min.
Removing a Value
- Removing the maximum decreases the range (and may decrease the mean).
- Removing the minimum increases the minimum, so range decreases (and may increase the mean).
- Removing a value equal to the mean leaves the mean unchanged.
Changing Every Value by a Constant
| Operation | Effect on Mean | Effect on Median | Effect on Range |
|---|---|---|---|
| Add k to every value | Mean + k | Median + k | No change |
| Subtract k from every value | Mean − k | Median − k | No change |
| Multiply every value by k | Mean × k | Median × k | Range × k |
| Divide every value by k | Mean / k | Median / k | Range / k |
Key insight: Adding or subtracting a constant shifts all values by the same amount — so the mean and median shift by the same amount, but the range (a difference) stays the same. Multiplying scales all values and the range equally.
Average Rate of Change
The average rate of change measures how much a quantity changes per unit of another quantity — typically how a value changes over time, distance, or another variable. It is the slope formula applied to real-world contexts.
= (y₂ − y₁) / (x₂ − x₁)
// This is also the slope formula — rise over run
The Average Speed Trap — Two Legs of a Journey
When a trip has two legs at different speeds, the average speed is NOT the arithmetic average of the two speeds. You must use: Total distance ÷ Total time.
The 30 mph leg takes twice as long as the 60 mph leg (2 hours vs. 1 hour). Alicia spends more time driving slowly, so the slow speed has more influence on the average. The harmonic-mean intuition: average speed is always pulled toward the slower speed when equal distances are covered.
Test Day Strategy
Top ACT Traps to Avoid
- Average of averages: Never add group averages and divide — always go back to sums.
- Average speed: For two-leg journeys, use Total distance ÷ Total time, never (v₁ + v₂)/2.
- Forgetting to sort before finding the median — data on the ACT is almost never pre-sorted.
- Even vs. odd count for median — always check whether n is even (average two middle values) or odd (one middle value).
- Range only uses max and min — don't accidentally calculate something else.
- Adding a value equal to the mean doesn't change the mean — know this cold for true/false style questions.
Problem-Type Decision Map
- "Find the missing value given the average" → Sum = Mean × Count, subtract known sum.
- "What score is needed to raise the average to X?" → Same approach: target sum minus current sum.
- "Combined average of two groups" → Compute each sum, add them, divide by total count.
- "What is the average speed for the whole trip?" → Total distance ÷ Total time.
- "Effect of adding/removing a value" → Use new sum ÷ new count; re-sort for median.
- "Weighted average" → Σ(value × weight) ÷ Σ(weights).
Quick Reference — All Formulas
| Concept | Formula | Key Watch-out |
|---|---|---|
| Mean | Mean = Sum / Count | Convert mean → sum as first step |
| Sum (from mean) | Sum = Mean × Count | Use this to find missing values |
| Median (odd n) | Position = (n+1)/2 | Must sort data first |
| Median (even n) | Average positions n/2 and n/2+1 | Average the two middle values |
| Mode | Most frequent value | Can be none, one, or many |
| Range | Max − Min | Only two values needed |
| Weighted mean | Σ(v × w) / Σ(w) | Weights must sum to total weight |
| Combined mean | (Sum₁ + Sum₂)/(n₁ + n₂) | Never average the averages |
| Average rate of change | (y₂ − y₁)/(x₂ − x₁) | Same as slope formula |
| Average speed | Total distance / Total time | Compute time for each leg first |
Put It to the Test
You've covered every averages concept the ACT tests. Now build speed and accuracy with ACT-style questions. Start with Quiz 1 and progress through Quiz 3 — difficulty increases with each set.