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Calculate measures of central tendency
Learn step-by-step with practice exercises built right in.
How do we describe the "center" or "typical value" of a data set? These three measures help us understand data and make comparisons!
Measures of central tendency describe the center or typical value of a data set.
Three main measures:
Each tells us something different about the data!
The mean is the arithmetic average of all values.
Formula: Mean = Sum of all values ÷ Number of values
Symbol: x̄ (read as "x bar")
Process:
Find the mean of: 8, 12, 15, 10, 5
Step 1: Add all numbers. 8 + 12 + 15 + 10 + 5 = 50
Step 2: Count how many numbers. 5 numbers
Step 3: Divide sum by count. Mean = 50 ÷ 5 = 10
Answer: Mean = 10
Find the median of: 3, 7, 2, 9, 5
Avoid these 3 frequent errors
See how this math is used in the real world
Solve .
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The mean is what people usually mean by "average"!
Example: Test scores: 85, 90, 78, 92, 85
Step 1: Add all values 85 + 90 + 78 + 92 + 85 = 430
Step 2: Count how many values 5 scores
Step 3: Divide Mean = 430 ÷ 5 = 86
The mean score is 86!
Example: Ages: 12, 15, 13, 11, 14, 15
Sum = 12 + 15 + 13 + 11 + 14 + 15 = 80 Count = 6 Mean = 80 ÷ 6 ≈ 13.33 years
Mean can be a decimal even if all data are whole numbers!
The median is the middle value when data is arranged in order.
Process:
The median divides the data in half!
Example: 7, 3, 9, 5, 1
Step 1: Arrange in order 1, 3, 5, 7, 9
Step 2: Find the middle 1, 3, 5, 7, 9
Median = 5 (middle value)
With odd number of values, there's one middle value!
Example: 8, 4, 6, 2
Step 1: Arrange in order 2, 4, 6, 8
Step 2: Find the two middle values 2, 4, 6, 8
Step 3: Average the two middle values Median = (4 + 6) ÷ 2 = 10 ÷ 2 = 5
Median = 5
With even number of values, average the two middle values!
The mode is the value that appears most frequently.
Example: 5, 7, 3, 7, 9, 7, 4
7 appears three times (more than any other)
Mode = 7
The mode is the most common value!
Some data sets have no mode:
Example: 5, 8, 11, 14, 17
Each value appears only once.
No mode (or all values are modes)
This is perfectly fine!
Data can have more than one mode:
Example: 3, 5, 5, 7, 9, 9, 11
Both 5 and 9 appear twice.
Modes = 5 and 9 (bimodal)
Bimodal = two modes Multimodal = more than two modes
Data: 2, 4, 4, 5, 7, 9
Mean: (2 + 4 + 4 + 5 + 7 + 9) ÷ 6 = 31 ÷ 6 ≈ 5.17
Median: 2, 4, 4, 5, 7, 9 → (4 + 5) ÷ 2 = 4.5
Mode: 4 (appears twice)
All different! Each measures "center" differently.
Mean:
Median:
Mode:
Outlier: A value much larger or smaller than others
Example: Salaries: 32,000, 33,000, $200,000
Mean: (32,000 + 33,000 + 330,000 ÷ 5 = $66,000
Median: 32,000, **35,000, 33,000
The outlier (33,000) better represents typical salary.
In symmetric data, mean ≈ median:
Example: 10, 12, 14, 16, 18
Mean = 70 ÷ 5 = 14 Median = 14 (middle value)
When distribution is balanced, mean and median are similar!
Right-skewed (positive skew): Few large values
Left-skewed (negative skew): Few small values
Median is more resistant to skewness!
Class test scores: 65, 70, 75, 75, 80, 80, 80, 85, 90, 95
Mean: 795 ÷ 10 = 79.5
Median: 65, 70, 75, 75, 80, 80, 80, 85, 90, 95 = (80 + 80) ÷ 2 = 80
Mode: 80 (appears three times)
Most students scored around 80!
Home prices in neighborhood: 160K, 175K, 900K
Mean: 289K
Median: 160K, 175K, 900K = (175K) ÷ 2 = $172.5K
Median (289K) inflated by the $900K mansion.
Problem: Four numbers have mean of 15. Three numbers are 12, 14, and 18. Find the fourth.
Solution: Mean = 15, Count = 4 Sum of all four = 15 × 4 = 60
Known sum: 12 + 14 + 18 = 44 Missing value: 60 - 44 = 16
The fourth number is 16!
Mode works great for non-numerical data:
Favorite colors: Blue, Red, Blue, Green, Blue, Yellow, Red, Blue
Mode = Blue (appears 4 times)
Can't calculate mean or median for colors!
Mode is the only measure that works for categorical data.
Example: 10, 10, 10, 10, 10
Mean: 50 ÷ 5 = 10 Median: 10 (middle value) Mode: 10 (appears five times)
When all values are the same, all three measures equal that value!
Original: 5, 7, 9 (Mean = 7, Median = 7)
Add 11: New data: 5, 7, 9, 11
New mean: 32 ÷ 4 = 8 New median: (7 + 9) ÷ 2 = 8
Adding values changes mean and median!
Range measures spread, not center:
Range = Largest value - Smallest value
Example: 3, 7, 9, 12, 14
Range = 14 - 3 = 11
Range tells how spread out the data is!
Data: 15, 12, 18, 12, 20, 15, 12
Mean: Sum = 15 + 12 + 18 + 12 + 20 + 15 + 12 = 104 Count = 7 Mean = 104 ÷ 7 ≈ 14.86
Median: Order: 12, 12, 12, 15, 15, 18, 20 Middle (4th value) = 15
Mode: 12 appears three times (most frequent) Mode = 12
When data is already grouped:
Example:
Calculate: (70×2 + 80×5 + 90×3) ÷ (2 + 5 + 3) = (140 + 400 + 270) ÷ 10 = 810 ÷ 10 = 81
Mean score = 81
If mean = 12 and there are 8 values:
Sum of all values = Mean × Count = 12 × 8 = 96
Useful for reverse problems!
Different values have different importance:
Example: Final grade
Weighted mean: (80×3 + 85×3 + 90×3 + 95×1) ÷ (3 + 3 + 3 + 1) = (240 + 255 + 270 + 95) ÷ 10 = 860 ÷ 10 = 86
Some values count more!
❌ Mistake 1: Not ordering data for median
❌ Mistake 2: Forgetting to average two middle values
❌ Mistake 3: Confusing mode with median
❌ Mistake 4: Thinking there must be a mode
❌ Mistake 5: Rounding too early
Use mean when:
Use median when:
Use mode when:
Context determines the best choice!
Education:
Business:
Sports:
Economics:
To find mean:
To find median:
To find mode:
To choose which to use:
Mean (Average):
Median (Middle):
Mode (Most Common):
Remember:
Tip 1: Always organize data first
Tip 2: Check your work
Tip 3: Consider outliers
Tip 4: Practice with real data
Tip 5: Understand what each measures
Three measures describe the center of data:
Mean:
Median:
Mode:
Key concepts:
Applications:
Problem-solving:
Understanding mean, median, and mode is fundamental for data analysis and statistics!
Step 1: Put in order from least to greatest. 2, 3, 5, 7, 9
Step 2: Find the middle number. 5 numbers, so the 3rd number is in the middle.
Step 3: Identify the median. The middle value is 5.
Answer: Median = 5
Find the mode of: 4, 7, 2, 7, 9, 7, 3
Step 1: Count how many times each number appears. 2: once 3: once 4: once 7: three times 9: once
Step 2: Identify the most frequent. 7 appears most often (3 times)
Answer: Mode = 7
Find the mean, median, and mode of: 12, 15, 11, 15, 13, 10, 15, 14
Mean: Sum = 12 + 15 + 11 + 15 + 13 + 10 + 15 + 14 = 105 Count = 8 numbers Mean = 105 ÷ 8 = 13.125
Median: Ordered: 10, 11, 12, 13, 14, 15, 15, 15 8 numbers (even), so average the 4th and 5th. Middle values: 13 and 14 Median = (13 + 14) ÷ 2 = 13.5
Mode: 15 appears 3 times (most frequent) Mode = 15
Answer: Mean = 13.125, Median = 13.5, Mode = 15
Test scores are: 85, 90, 78, 92, 88, 95, 45. Which measure of center (mean, median, or mode) best represents the typical score? Explain why.
Step 1: Calculate mean. Sum = 85 + 90 + 78 + 92 + 88 + 95 + 45 = 573 Mean = 573 ÷ 7 = 81.86
Step 2: Find median. Ordered: 45, 78, 85, 88, 90, 92, 95 Median = 88 (middle value)
Step 3: Find mode. All numbers appear once, no mode.
Step 4: Analyze. 45 is an OUTLIER (much lower than others) The mean (81.86) is pulled down by the outlier The median (88) better represents typical score Most scores are in the 78-95 range
Answer: The MEDIAN (88) best represents the typical score because the mean is affected by the outlier score of 45.