Session 12: Statistics — Mean, Median, Mode & Range

Explanatory SVG

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1. Mean (Average)

The mean is what most people call the "average." To find it, add all the values together and divide by the number of values.

\[\text{Mean} = \frac{\text{sum of all values}}{\text{number of values}}\]

Example 1: A student scored 78, 85, 90, 72, and 95 on five tests. What is the mean score?

\[\text{Mean} = \frac{78 + 85 + 90 + 72 + 95}{5} = \frac{420}{5} = 84\]

The mean score is 84.

Example 2: Four employees earn $3,200, $4,100, $2,800, and $3,900 per month. What is the mean monthly salary?

\[\text{Mean} = \frac{3200 + 4100 + 2800 + 3900}{4} = \frac{14000}{4} = 3500\]

The mean salary is $3,500.

Watch out: Always divide by the number of values, not by the range, and not by the largest value. Count how many numbers are in your data set carefully — this is where many errors happen.

Real-life uses of the mean include grade point averages, average rainfall in a city, average daily temperature, and average cost per unit.

2. Median (Middle Value)

The median is the middle value when the data is arranged in order from least to greatest. It's the measure that cuts the data set exactly in half.

• If there is an odd number of values, the median is the single middle value.
• If there is an even number of values, the median is the average of the two middle values.

Example (odd count): Find the median of 3, 7, 9, 12, 15.

Already in order. There are 5 values, so the middle is the 3rd value: 9.

Example (even count): Find the median of 4, 8, 11, 15.

There are 4 values. The two middle values are 8 and 11.

\[\text{Median} = \frac{8 + 11}{2} = \frac{19}{2} = 9.5\]

When is the median better than the mean? When the data has extreme values called outliers. For example, if four friends earn $40,000 per year but one earns $2,000,000, the mean salary looks like ~$435,000 — which doesn't represent anyone. The median salary of $40,000 is far more accurate. This is why you often hear about "median household income" in news reports rather than mean income.

Real-life: median home prices, median salaries, median test scores in a class.

3. Mode

The mode is the value that appears most often in a data set. A data set can have:

• No mode — if all values appear the same number of times
• One mode — one value appears more than all others
• Two modes (bimodal) — two values tie for most frequent
• Multiple modes — more than two values tie

Example 1 (one mode): 2, 4, 4, 7, 9, 4 → mode = 4 (appears 3 times)

Example 2 (no mode): 1, 2, 3, 4 → no mode (each appears once)

Example 3 (bimodal): 2, 2, 5, 5, 8 → modes = 2 and 5 (each appears twice)

GED® Tip: The mode is especially useful in business and consumer contexts. A shoe store wants to know the most common shoe size (mode, not mean). A school might track the most frequent score on a test to see where most students landed.

4. Range

The range measures how spread out a data set is. It's simply the difference between the largest and smallest values.

\[\text{Range} = \text{Maximum} - \text{Minimum}\]

Example: Data set: 12, 5, 18, 9, 23, 7

• Maximum = 23, Minimum = 5
• Range = 23 − 5 = 18

A small range tells you the data is clustered together (consistent). A large range tells you the data is spread out (inconsistent). For example, a basketball player who scores 18, 19, 20, 18, 21 points (range = 3) is very consistent. A player who scores 3, 28, 5, 31, 8 points (range = 28) is unpredictable.

5. Interpreting Data in Context

On the GED®, you'll often be given a table or list of numbers and asked to compute one or more of these statistics. Let's practice with a full word problem.

Word Problem 1: A teacher records quiz scores for 6 students: 80, 92, 75, 88, 80, 76.

• Mean: (80+92+75+88+80+76) ÷ 6 = 491 ÷ 6 ≈ 81.8
• Median: Ordered: 75, 76, 80, 80, 88, 92. Two middle values: 80 and 80 → median = 80
• Mode: 80 appears twice; mode = 80
• Range: 92 − 75 = 17

Word Problem 2: Monthly rainfall (inches): 2, 4, 3, 7, 5, 4, 6, 4, 3, 5, 2, 8

• Mean = (2+4+3+7+5+4+6+4+3+5+2+8) ÷ 12 = 53 ÷ 12 ≈ 4.4 inches
• Mode = 4 (appears 3 times)
• Range = 8 − 2 = 6 inches

Notice how the mean, median, mode, and range each tell you something different about the data. Together, they give you a complete picture.

Comparing Two Data Sets: If Set A has mean 70 and range 5, and Set B has mean 70 and range 30, both groups perform similarly on average, but Set B is far less consistent.