25-30 minutes
Lesson 1: Measures of Center
Learning Objectives
By the end of this lesson, you will be able to:
- Define and calculate the mean (average) of a dataset
- Find the median (middle value) for both odd and even datasets
- Identify the mode (most frequent value) and recognize when there's no mode, one mode, or multiple modes
- Understand the properties of each measure and when each is affected by outliers
- Choose the most appropriate measure of center based on data characteristics
What Are Measures of Center?
Imagine you want to describe a dataset with just one number that represents the "typical" or "center" value. That's what measures of center do! They summarize an entire dataset into a single value that gives you a sense of where the "middle" is.
The three most common measures of center are:
- Mean - The arithmetic average
- Median - The middle value when data is ordered
- Mode - The most frequently occurring value
Each measure tells a slightly different story about your data, and choosing the right one depends on your data's characteristics!
The Mean (Average)
Mean (x̄): The sum of all values divided by the number of values. Also called the "average" or "arithmetic mean."
Formula:
Mean = (Sum of all values) ÷ (Number of values)
or symbolically:
x̄ = Σx / n
(Σ means "sum of", x represents each value, n = number of values)
Example 1: Calculating the Mean
Dataset: Test scores for 5 students: 72, 85, 90, 78, 85
Step-by-step calculation:
- Add all values: 72 + 85 + 90 + 78 + 85 = 410
- Count how many values: There are 5 test scores
- Divide sum by count: 410 ÷ 5 = 82
Answer: The mean test score is 82
Key Properties of the Mean
- Uses all data values - Every number in the dataset affects the mean
- Sensitive to outliers - Extreme values can pull the mean up or down significantly
- Unique - There's only one mean for any dataset
- Balance point - The mean is the value where the data "balances"
Watch Out: The Mean Can Be Misleading!
Imagine 5 employees at a small company earn: $30k, $32k, $35k, $38k, and the CEO earns $500k.
Mean salary: ($30k + $32k + $35k + $38k + $500k) ÷ 5 = $127k
Does $127k represent a "typical" employee salary? No! Four out of five employees earn under $40k. The CEO's extreme salary pulled the mean way up. This is why the median is often better for skewed data!
The Median (Middle Value)
Median: The middle value when the dataset is arranged in order from smallest to largest. Half the values are above the median, half are below.
How to Find the Median:
- Order the data from smallest to largest
- If n is odd: The median is the middle value
- If n is even: The median is the average of the two middle values
Example 2A: Median with ODD number of values
Dataset: Ages of 7 children: 5, 8, 6, 10, 7, 9, 6
Step-by-step:
- Order the data: 5, 6, 6, 7, 8, 9, 10
- Count: 7 values (odd number)
- Find the middle: The 4th value is in the middle (3 values before it, 3 after)
Answer: The median age is 7
Example 2B: Median with EVEN number of values
Dataset: Hours studied by 6 students: 2, 5, 3, 7, 4, 6
Step-by-step:
- Order the data: 2, 3, 4, 5, 6, 7
- Count: 6 values (even number)
- Find the two middle values: 4 and 5 (both highlighted)
- Average them: (4 + 5) ÷ 2 = 4.5
Answer: The median is 4.5 hours
Key Properties of the Median
- Resistant to outliers - Extreme values don't affect the median much
- Represents the "middle" - Literally! 50% of data is above, 50% below
- Better for skewed data - When you have outliers, median is often more representative than mean
- Easy to understand - "The middle value" is intuitive
Why Median Beats Mean (Sometimes)
Let's revisit that company salary example:
Salaries: $30k, $32k, $35k, $38k, $500k
Already ordered! With 5 values (odd), the median is the 3rd value: $35k
Compare:
- Mean: $127k (misleading – only the CEO earns that much!)
- Median: $35k (much better representation of a "typical" salary)
This is why you often hear about "median household income" in economic reports – it's not affected by billionaires!
The Mode (Most Frequent Value)
Mode: The value that appears most frequently in a dataset. A dataset can have no mode, one mode, or multiple modes.
Example 3A: One Mode (Unimodal)
Dataset: Shoe sizes: 7, 8, 8, 9, 8, 10, 11, 8
The number 8 appears 4 times (more than any other value).
Mode = 8
Example 3B: Two Modes (Bimodal)
Dataset: Test scores: 70, 75, 80, 80, 85, 90, 90, 95
Both 80 and 90 appear twice (tied for most frequent).
Modes = 80 and 90 (bimodal)
Example 3C: No Mode
Dataset: Ages: 18, 19, 20, 21, 22
Every value appears exactly once – no value is "most frequent."
No mode
Key Properties of the Mode
- Works for any data type - Can find mode of numbers, categories, even text!
- Not unique - Can have multiple modes (or no mode at all)
- Most useful for categorical data - "What's the most popular ice cream flavor?"
- Not affected by outliers - Only frequency matters, not extreme values
Mode with Categorical Data
Mode is especially useful when your data isn't numerical!
Survey: Favorite ice cream flavors from 20 people:
Chocolate (8), Vanilla (5), Strawberry (4), Mint (3)
Mode = Chocolate (most popular choice)
You can't calculate a mean or median of "chocolate" and "vanilla" – but you CAN find the mode!
Comparing the Three Measures
| Measure | Definition | When to Use | Affected by Outliers? |
|---|---|---|---|
| Mean | Sum ÷ Count | Symmetric data with no extreme outliers | YES - Very sensitive |
| Median | Middle value | Skewed data or data with outliers | NO - Resistant |
| Mode | Most frequent | Categorical data or finding most common value | NO - Not affected |
All Three Together: Test Scores Example
Dataset: 10 students' test scores: 65, 70, 75, 75, 80, 85, 85, 85, 90, 95
Mean:
Sum = 65+70+75+75+80+85+85+85+90+95 = 805
Mean = 805 ÷ 10 = 80.5
Median:
10 values (even), so average the 5th and 6th values: (80 + 85) ÷ 2 = 82.5
Mode:
85 appears three times (more than any other score) = 85
Summary: Mean = 80.5, Median = 82.5, Mode = 85. All three are close together because this data is fairly symmetric!
Check Your Understanding
Test your knowledge with these practice questions!
Question 1:
Calculate the mean of: 12, 15, 18, 21, 24
Question 2:
Find the median of: 3, 9, 5, 12, 7, 6, 8
Question 3:
What is the mode of: Red, Blue, Red, Green, Blue, Red, Yellow
Question 4:
Home prices in a neighborhood: $200k, $220k, $210k, $205k, $2.5M (one mansion). Which measure best represents a "typical" home price?
Question 5:
Which measure of center uses EVERY value in the dataset in its calculation?
Key Takeaways
- Mean = Add all values, divide by count. Sensitive to outliers.
- Median = Middle value when ordered. Resistant to outliers – use for skewed data!
- Mode = Most frequent value. Works for categorical data. Can have 0, 1, or multiple modes.
- When in doubt with outliers: Use median instead of mean!
- All three matter: Looking at mean AND median together tells you about data shape!