Measures of Center

Welcome to measures of center! These important statistics help us describe a set of data with a single representative number. They answer the question: "What's typical for this data?"

What Are Measures of Center?

Measures of center (also called measures of central tendency) are single numbers that represent the "middle" or "typical" value in a data set. The three main measures of center are mean, median, and mode.

Mean (Average)

The mean is what most people call the "average." It's the sum of all the values divided by how many values there are.

How to Find the Mean

Step 1: Add up all the numbers in the data set

Step 2: Count how many numbers there are

Step 3: Divide the sum by the count

Formula: Mean = Sum of all values ÷ Number of values

Example 1: Finding the Mean

Test scores: 85, 90, 78, 92, 85

Step 1: Add all the scores 85 + 90 + 78 + 92 + 85 = 430

Step 2: Count the scores There are 5 scores

Step 3: Divide 430 ÷ 5 = 86

The mean is 86

Example 2: Mean with Different Values

Hours of sleep: 8, 7, 9, 7, 8, 6, 9

Sum: 8 + 7 + 9 + 7 + 8 + 6 + 9 = 54 Count: 7 nights Mean: 54 ÷ 7 = 7.71 hours (about 7.7 hours)

When to Use Mean

The mean is useful when:

• You want to know the typical value across all data points
• The data doesn't have extreme outliers
• You need to consider every value in the data set

Median (Middle Value)

The median is the middle number when the data is arranged in order from least to greatest.

How to Find the Median

Step 1: Arrange all numbers in order from smallest to largest

Step 2: Find the middle number

• If there's an odd number of values, the median is the middle one
• If there's an even number of values, the median is the average of the two middle numbers

Example 1: Odd Number of Values

Ages: 12, 15, 13, 14, 11

Step 1: Put in order 11, 12, 13, 14, 15

Step 2: Find the middle 11, 12, 13, 14, 15

The median is 13 (the 3rd value out of 5)

Example 2: Even Number of Values

Prices: 5, 8, 6, 9, 7, 10

Step 1: Put in order 5, 6, 7, 8, 9, 10

Step 2: Find the two middle numbers 5, 6, 7, 8, 9, 10

Step 3: Average the two middle numbers (7 + 8) ÷ 2 = 15 ÷ 2 = 7.5

The median is 7.5

When to Use Median

The median is useful when:

• The data has outliers (extreme values)
• You want a measure that isn't affected by very high or very low values
• You need to find the exact middle of the data

Mode (Most Frequent)

The mode is the number that appears most often in a data set.

How to Find the Mode

Step 1: Look at all the numbers

Step 2: Count how many times each number appears

Step 3: The number that appears most frequently is the mode

Example 1: One Mode

Shoe sizes: 7, 8, 7, 9, 7, 6, 8

Count each size:

• Size 6: appears 1 time
• Size 7: appears 3 times ← Most frequent
• Size 8: appears 2 times
• Size 9: appears 1 time

The mode is 7

Example 2: Multiple Modes (Bimodal)

Test scores: 85, 90, 85, 78, 90, 88

Count each score:

• 78: appears 1 time
• 85: appears 2 times ← Tied for most
• 88: appears 1 time
• 90: appears 2 times ← Tied for most

The modes are 85 and 90 (this is called bimodal)

Example 3: No Mode

Numbers: 5, 7, 9, 11, 13

Each number appears only once, so there is no mode.

When to Use Mode

The mode is useful when:

• You want to know what's most common
• You're working with categories (like favorite colors)
• You want a measure that's actually in the data set

Comparing Mean, Median, and Mode

Let's look at the same data set with all three measures:

Data: 5, 7, 8, 8, 10, 12, 35

Mean: Sum = 5 + 7 + 8 + 8 + 10 + 12 + 35 = 85 Count = 7 Mean = 85 ÷ 7 ≈ 12.14

Median: Data is already in order: 5, 7, 8, 8, 10, 12, 35 Median = 8 (middle value)

Mode: 8 appears twice, all others appear once Mode = 8

Notice: The mean (12.14) is much higher than the median (8) because of the outlier 35. The median and mode give a better sense of the "typical" value in this case.

Outliers and Their Effect

An outlier is a value that is much higher or much lower than the other values in the data set.

Example: Test scores are 85, 88, 90, 87, 89, 20

The score of 20 is an outlier (much lower than the others).

Effect on measures:

• Mean: Very affected by outliers (pulls the average down)
• Median: Not much affected by outliers
• Mode: Not affected by outliers (unless the outlier appears most often)

Real-World Applications

Mean:

• Calculating your grade point average
• Finding average temperature for a month
• Determining average points per game in sports

Median:

• Median home prices (because a few very expensive homes can skew the mean)
• Median income (better than mean when there are very wealthy outliers)
• Median age in a population

Mode:

• Most popular shoe size to stock in a store
• Most common favorite color in a class
• Most frequent number of siblings

Choosing the Right Measure

Use the mean when:

• Data is fairly evenly distributed
• No extreme outliers exist
• You need to use all the data values

Use the median when:

• Data has outliers
• You want the true middle value
• Data is skewed (lopsided)

Use the mode when:

• You want to know what's most common
• Working with categories or non-numerical data
• You need a value that actually appears in the data

Common Mistakes to Avoid

1. Forgetting to order the data for median: Always arrange from least to greatest first

2. Dividing by the wrong number for mean: Count the values carefully

3. Not averaging the two middle numbers: When there's an even number of values, you must average the two middle ones for median

4. Calling the mode the middle number: Mode is most frequent, not the middle

5. Rounding too early: Keep extra decimal places in your work and round only at the end

6. Using mean when outliers exist: Consider median instead when you have extreme values

Practice Tips

To master measures of center:

For Mean:

• Practice with different sized data sets
• Use a calculator for larger numbers
• Check your work by estimating (should be between the smallest and largest values)

For Median:

• Always write the numbers in order first
• Use your finger to cross off numbers from both ends until you reach the middle
• Remember to average the two middle numbers for even-sized data sets

For Mode:

• Make a tally chart to count frequencies
• Remember a data set can have 0, 1, or more modes
• When graphing, the mode is the tallest bar or highest point

Advanced Concepts

Range: The difference between the highest and lowest values

• Tells you how spread out the data is
• Range = Maximum - Minimum

Using all three together:

• If mean, median, and mode are close together, the data is fairly symmetric
• If they're far apart, the data may be skewed or have outliers
• Comparing all three gives the most complete picture

Data Sets Practice

Try finding mean, median, and mode for these:

Set 1: 10, 15, 12, 18, 15, 20, 15

• Mean = (10+15+12+18+15+20+15) ÷ 7 = 105 ÷ 7 = 15
• Median = 10, 12, 15, 15, 15, 18, 20 → 15
• Mode = 15 (appears 3 times)

Set 2: 5, 8, 12, 15, 20, 100

• Mean = (5+8+12+15+20+100) ÷ 6 = 160 ÷ 6 ≈ 26.67
• Median = 5, 8, 12, 15, 20, 100 → (12+15)÷2 = 13.5
• Mode = No mode (all appear once)

Notice how the outlier (100) affects the mean much more than the median!

Understanding measures of center helps you describe data accurately, make predictions, and understand statistics you encounter in news, sports, and everyday life. These skills form the foundation for more advanced statistics in middle school and beyond!