Measures of Center: Mean, Median, and Mode - Sixth Grade Mathematics
Measures of center are single representative numerical values that describe the central tendency or typical value of a statistical data set.
The Three Core Measures of Center
To summarize a large set of numbers with a single value, mathematicians use three distinct metrics, each offering a unique perspective on "centrality."
The Three Measures:
+-----------+-----------------------------------------------+-----------------------------+
| Measure | Definition | How to Calculate |
+-----------+-----------------------------------------------+-----------------------------+
| Mean | The arithmetic balance point (average); | Sum all values and divide |
| | the value if all amounts were shared equally. | by the total number of items|
+-----------+-----------------------------------------------+-----------------------------+
| Median | The exact middle value when data is sorted in | Order numbers; pick middle |
| | ascending order; splits data 50/50. | (or average 2 middle values)|
+-----------+-----------------------------------------------+-----------------------------+
| Mode | The value(s) that appear most frequently; | Count frequencies; can have |
| | the most popular observation. | none, one, or multiple modes|
+-----------+-----------------------------------------------+-----------------------------+
The Mean as a Fair Share (Balance Point)
Conceptually, the mean represents the "fair share" outcome. If five friends pool all their money together and distribute it so everyone receives an equal amount, each friend gets the mean.
Formula for the Mean:
Mean = (Sum of all data values) / (Total number of data values)
Mean = (Sigma x) / N
Example: Test scores: 80, 85, 90, 75, 95
Sum = 80 + 85 + 90 + 75 + 95 = 425
Count (N) = 5
Mean = 425 / 5 = 85.
Physically, if you place weights on a ruler at each data point, the mean is the exact fulcrum position where the ruler balances horizontally.
Finding the Median: Odd vs. Even Sample Sizes
The median is strictly positional. Data MUST be sorted from smallest to largest first!
Case 1: Odd Count of Data Points (Unique Middle)
Data: 14, 7, 12, 20, 9
Step 1: Sort: 7, 9, [12], 14, 20
Step 2: Locate exact middle: 12.
Median = 12.
Case 2: Even Count of Data Points (Two Middle Numbers)
Data: 8, 15, 3, 11, 22, 19
Step 1: Sort: 3, 8, [11, 15], 19, 22
Step 2: There are two middle numbers (11 and 15).
Step 3: Calculate the mean of the two middle numbers:
Median = (11 + 15) / 2 = 26 / 2 = 13.
Median = 13.
Notice that with an even count, the median (13) does not even have to be a number in the original data set!
The Mode
The mode is the most frequently occurring value in the set.
A data set can have:
- One mode (unimodal): e.g., in 2, 4, 4, 7, 9, the mode is 4.
- Two modes (bimodal): e.g., in 3, 3, 5, 8, 8, 10, the modes are 3 and 8.
- Multiple modes (multimodal).
- No mode: if every value occurs with the exact same frequency (e.g., 5, 8, 12, 17, 21), there is NO mode.
The mode is the only measure of center that can be used for categorical data (e.g., most popular pizza topping).
Choosing the Best Measure of Center
The choice of which measure to report depends upon the presence of extreme values (outliers) and distribution shape: The Mean incorporates every value, making it ideal for symmetrical data without outliers. However, a single extreme outlier pulls the mean dramatically toward it. The Median relies solely on position and is resistant to outliers. When data is skewed or contains outliers (like household incomes or home prices), the median provides a much more accurate reflection of the typical value. The Mode is best for popularity contests, stocking retail inventory, or categorical data.
Exercises
Exercise 1. Calculate the mean, median, and mode for each data set: (a) 4, 8, 6, 9, 8; (b) 12, 15, 11, 19, 14, 13; (c) 100, 2, 5, 3, 5.
Exercise 2. A student has scores of 82, 88, and 94 on three science exams. What score must the student earn on the fourth exam to achieve an overall mean score of 90?
Exercise 3. The weekly wages of five employees at a small bakery are $400, $420, $410, $430, and $2,000 (the owner). (a) Calculate the mean weekly wage; (b) Calculate the median weekly wage; (c) Which measure better represents the typical employee wage, and why?
Exercise 4. A set of seven distinct numbers has a median of 25. If the three lowest numbers are 10, 15, and 18, could the number 30 be one of the numbers? Explain.
Solutions
Solution to Exercise 1. For (a), Sum = 4 + 8 + 6 + 9 + 8 = 35. Mean = 35 / 5 = 7. Ordered: 4, 6, [8], 8, 9. Median = 8. Mode = 8 (appears twice). For (b), Sum = 12 + 15 + 11 + 19 + 14 + 13 = 84. Mean = 84 / 6 = 14. Ordered: 11, 12, [13, 14], 15, 19. Two middle values 13 and 14: Median = (13 + 14)/2 = 13.5. Mode = None (all values appear once). For (c), Sum = 100 + 2 + 5 + 3 + 5 = 115. Mean = 115 / 5 = 23. Ordered: 2, 3, [5], 5, 100. Median = 5. Mode = 5.
Solution to Exercise 2. Let x be the fourth exam score. To achieve a mean of 90 across 4 exams, the total sum of all 4 scores must equal 4 * 90 = 360 points. Sum of first 3 scores = 82 + 88 + 94 = 264. Equation: 264 + x = 360. Subtract 264: x = 360 - 264 = 96. The student must score 96 on the fourth exam.
Solution to Exercise 3. For (a), Mean = (400 + 420 + 410 + 430 + 2000) / 5 = 3660 / 5 = $732. For (b), Ordered wages: $400, $410, [$420], $430, $2000. Median = $420. For (c), the median ($420) is a far better representation of a typical worker's pay. The owner's $2,000 wage is an extreme outlier that inflates the mean to $732, an amount that no regular employee actually earns!
Solution to Exercise 4. Yes. In an ordered set of 7 numbers, the median is the 4th value: Position 1 (10), Position 2 (15), Position 3 (18), Position 4 (25). Positions 5, 6, and 7 must simply be values greater than 25. Since 30 is greater than 25, it can easily occupy Position 5, 6, or 7.