9.6Unit Review — One-Variable Statistics
Review all Unit 9 concepts: measures of center, measures of spread, dot plots, histograms, box plots, five-number summary, and comparing distributions.
Why This Matters
Statistics is one of the fastest-growing fields in math and science. A strong review of Unit 9 prepares you for two-variable statistics in Unit 10, AP Statistics, and data analysis skills that are valuable in virtually every career.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Unit Overview
Unit 9 — One-Variable Statistics introduces the tools statisticians use to analyze a single set of data. We began by learning how to measure the center of a data set (mean, median, mode) and the spread (range, IQR, MAD). We then explored how to display data visually using dot plots, histograms, and box plots, and how to read the shape of a distribution. We learned to construct and interpret the five-number summary and identify outliers using the IQR fence method. Finally, we developed the skill of comparing two distributions — analyzing center, spread, shape, and outliers together to draw evidence-based conclusions.
Essential Question
How can measures of center, measures of spread, and graphical displays help us analyze and interpret real-world data?
Unit 9 Concept Map
Measures of Center
Mean · Median · Mode
Measures of Spread
Range · IQR · MAD
Dot Plots & Histograms
Shape · Clusters · Gaps
Box Plots
Five-Number Summary · Outliers
Comparing Distributions
Center · Spread · Shape
Unit 9 Learning Objectives — Review Checklist
☐ Calculate mean, median, and mode from a data set
☐ Identify which measure of center is most appropriate
☐ Calculate range, IQR, and MAD
☐ Explain what measures of spread tell you about data
☐ Read and create dot plots
☐ Read and create histograms
☐ Describe the shape of a distribution (symmetric, skewed, uniform, bimodal)
☐ Construct a five-number summary
☐ Draw and read a box plot
☐ Identify outliers using the IQR fence method
☐ Compare two data sets using center, spread, shape, and outliers
☐ Write a complete statistical conclusion with specific evidence
Vocabulary Review
Mean
The arithmetic average. Sum of all values ÷ number of values. Sensitive to outliers. Best for symmetric distributions.
Median
The middle value of an ordered data set. Resistant to outliers. Best for skewed distributions or data with outliers.
Mode
The value that appears most often. A data set can have no mode, one mode, or multiple modes.
Range
Max − Min. Measures total spread. Sensitive to outliers.
Quartile
Values that divide an ordered data set into four equal parts. Q1 = 25th percentile, Q2 = median = 50th percentile, Q3 = 75th percentile.
Interquartile Range (IQR)
Q3 − Q1. Measures the spread of the middle 50% of data. Resistant to outliers.
Mean Absolute Deviation (MAD)
The average distance of each data value from the mean. Measures typical spread around the mean.
Standard Deviation
A measure of spread that shows how far data values typically are from the mean. Larger SD = more spread. Related to MAD but uses squared differences.
Dot Plot
A graph where each data value is represented by a dot above a number line. Shows individual values, clusters, gaps, and outliers.
Histogram
A bar graph where each bar represents a range (interval) of values. The height of each bar shows the frequency. Shows shape and distribution of data.
Box Plot
A graph that displays the five-number summary. The box spans Q1 to Q3; a line marks the median; whiskers extend to Min and Max (excluding outliers).
Five-Number Summary
Min, Q1, Median (Q2), Q3, Max. Provides a complete picture of a data set's spread and center.
Outlier
A data value unusually far from the rest. Identified by: below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR). Shown as a separate point on a box plot.
Cluster
A group of data values that are close together in a dot plot or histogram, indicating a concentration of data in that region.
Gap
A region in a dot plot or histogram where no data values appear, indicating a break in the distribution.
Symmetric Distribution
A distribution where the left and right sides are mirror images. Mean ≈ Median. Box plot: median centered in box, equal whiskers.
Left-Skewed Distribution
A distribution with a long left tail. Most data is high; a few low values pull the mean below the median. Mean < Median.
Right-Skewed Distribution
A distribution with a long right tail. Most data is low; a few high values pull the mean above the median. Mean > Median.
Uniform Distribution
A distribution where all values occur with approximately equal frequency. The histogram has bars of roughly equal height.
Bimodal Distribution
A distribution with two distinct peaks (modes). The histogram has two tall bars separated by shorter bars.
Formula Review
Range
Range = Max − Min
Interquartile Range (IQR)
IQR = Q3 − Q1
Mean Absolute Deviation (MAD)
MAD = Σ|xᵢ − x̄| ÷ n
Average of all |value − mean| distances
Outlier Fences
Lower: Q1 − 1.5 × IQR Upper: Q3 + 1.5 × IQR
Values outside these fences are outliers
Five-Number Summary
1
Minimum
2
Q1 (25th %ile)
3
Median (Q2)
4
Q3 (75th %ile)
5
Maximum
Strategy Review
Choosing Measure of Center
- Mean: symmetric data, no outliers
- Median: skewed data or outliers present
- Mode: categorical data or most frequent value needed
Comparing Spread
- Larger IQR = more spread in middle 50%
- Larger range = more total spread
- Larger MAD = values farther from mean on average
- Use IQR (not range) when outliers are present
Reading Graphs
- Dot plot: count dots for frequency; find median by counting
- Histogram: bar height = frequency; look for shape and peak
- Box plot: box = IQR; line = median; whiskers = range (excluding outliers)
Identifying Outliers
- Step 1: Find Q1, Q3, IQR
- Step 2: Lower fence = Q1 − 1.5(IQR)
- Step 3: Upper fence = Q3 + 1.5(IQR)
- Step 4: Any value outside fences = outlier
Comparing Distributions — 4-Step Framework
Compare mean/median. Which is higher?
Compare IQR/range. Which is more variable?
Symmetric, skewed, or other? Same or different?
Any present? How do they affect the mean/range?
Visual Summary
Dot Plot
Histogram
Box Plot — Anatomy
Min
10
Q1
25
Median
40
Q3
55
Max
70
IQR = 55 − 25 = 30. Range = 70 − 10 = 60. Median (40) is centered in box → roughly symmetric.
Symmetric vs. Skewed Distributions
Symmetric
Mean ≈ Median. Equal whiskers.
Right-Skewed
Long right tail. Mean > Median.
Left-Skewed
Long left tail. Mean < Median.
Outlier Example
No Outliers — Range=40
With Outlier (75) — Range=65
Outlier check: IQR=20. Upper fence = 40 + 1.5(20) = 70. Value 75 > 70 → outlier. The outlier inflates the range (40→65) and pulls the mean upward, but the median and IQR are unchanged.
Review Examples
Find the mean, median, and mode of: 8, 12, 15, 12, 20, 18, 12, 25, 10
Order the data: 8, 10, 12, 12, 12, 15, 18, 20, 25
Mean = (8+10+12+12+12+15+18+20+25) ÷ 9 = 132 ÷ 9 ≈ 14.7
Median = middle value (5th of 9) = 12
Mode = 12 (appears 3 times)
Note: Mean (14.7) > Median (12) → slight right skew. The mode (12) is the most common value.
Find the range, IQR, and MAD for: 5, 10, 15, 20, 25
Range = 25 − 5 = 20
Ordered: 5, 10, 15, 20, 25. Median = 15. Q1 = 10, Q3 = 20.
IQR = Q3 − Q1 = 20 − 10 = 10
Mean = (5+10+15+20+25) ÷ 5 = 75 ÷ 5 = 15
Absolute deviations from mean: |5−15|=10, |10−15|=5, |15−15|=0, |20−15|=5, |25−15|=10
MAD = (10+5+0+5+10) ÷ 5 = 30 ÷ 5 = 6
Data: 14, 18, 21, 25, 28, 32, 35, 40, 45, 52. Construct the five-number summary and check for outliers.
Already ordered. n=10.
Min = 14, Max = 52
Median = (28+32)/2 = 30
Q1 = median of lower half {14,18,21,25,28} = 21
Q3 = median of upper half {32,35,40,45,52} = 40
Five-Number Summary: 14, 21, 30, 40, 52
IQR = 40 − 21 = 19
Lower fence = 21 − 1.5(19) = 21 − 28.5 = −7.5
Upper fence = 40 + 1.5(19) = 40 + 28.5 = 68.5
All values are between −7.5 and 68.5 → no outliers.
Data: 10, 12, 14, 15, 16, 18, 20, 22, 24, 80. Find the five-number summary and determine if 80 is an outlier.
Ordered: 10, 12, 14, 15, 16, 18, 20, 22, 24, 80. n=10.
Median = (16+18)/2 = 17
Q1 = median of {10,12,14,15,16} = 14
Q3 = median of {18,20,22,24,80} = 22
IQR = 22 − 14 = 8
Upper fence = 22 + 1.5(8) = 22 + 12 = 34
80 > 34 → 80 IS an outlier.
Five-Number Summary (with outlier noted): Min=10, Q1=14, Median=17, Q3=22, Max=80 (outlier).
A dot plot shows: 3, 4, 4, 5, 5, 5, 6, 6, 7, 9. Describe the distribution: shape, center, spread, and any outliers.
Ordered: 3, 4, 4, 5, 5, 5, 6, 6, 7, 9. n=10.
Center: Median = (5+5)/2 = 5. Mean = 54/10 = 5.4.
Spread: Range = 9−3 = 6. Q1=4, Q3=6, IQR=2.
Outlier check: Lower fence = 4−3=1; Upper fence = 6+3=9. Value 9 = upper fence exactly → borderline, not a clear outlier.
Shape: Cluster at 4–6. Slight right tail (value 9). Roughly symmetric with a slight right skew.
Description: The data clusters around 5 (median=5, mean=5.4). The distribution is roughly symmetric with a slight right skew. Range=6, IQR=2 — moderate spread.
A histogram shows: 0–9: 3, 10–19: 7, 20–29: 12, 30–39: 8, 40–49: 4, 50–59: 1. Describe the shape and estimate the center.
Total n = 3+7+12+8+4+1 = 35
Shape: Peaks in 20–29 range. More values on the left side of the peak than the right → slight right skew (tail extends right).
Estimated center: The peak interval is 20–29 (midpoint ≈ 24.5). The median is the 18th value. Counting: 3+7=10 (through 0–19), need 8 more → median falls in 20–29 interval. Estimated median ≈ 24.
Spread: Data ranges from 0 to 59 (range ≈ 59). Most data is in the 10–39 range.
Error Analysis — Find and Fix the Mistake
Error 1:
Data: 5, 8, 10, 12, 15. A student says: "The median is 10 and the IQR is 15 − 5 = 10."
Fix: The median is correct (10). But the student calculated the RANGE, not the IQR. IQR = Q3 − Q1 = 12 − 8 = 4.
Error 2:
A student says: "The distribution has Mean=80 and Median=65, so it is left-skewed."
Fix: When Mean > Median, the distribution is RIGHT-skewed (the right tail pulls the mean above the median). Left-skewed means Mean < Median.
Error 3:
Data: Q1=20, Q3=40, IQR=20. A student checks for outliers: "Upper fence = 40 + 1.5 = 41.5."
Fix: The student forgot to multiply by IQR. Upper fence = Q3 + 1.5 × IQR = 40 + 1.5(20) = 40 + 30 = 70.
Error 4:
A student compares two data sets and concludes: "Set A is better because it has a larger range."
Fix: A larger range means MORE variability, not better performance. The student should compare medians (for center/quality) and IQR (for consistency). A larger range is often undesirable.
Error 5:
Data: 4, 6, 8, 10, 12. A student says: "The MAD is (4+6+8+10+12)/5 = 8."
Fix: The student calculated the mean, not the MAD. MAD = average of |each value − mean|. Mean=8. Deviations: 4,2,0,2,4. MAD = (4+2+0+2+4)/5 = 12/5 = 2.4.
Common Mistakes
Using the mean when data has outliers — the mean gets pulled toward extreme values.
Use the median when data is skewed or has outliers. The median is resistant to extreme values.
Confusing IQR with range — using max − min when asked for the middle spread.
Range = max − min. IQR = Q3 − Q1. IQR measures the spread of the middle 50% of the data.
Leaving gaps between histogram bars — treating them like a bar chart.
Histogram bars represent continuous intervals and must touch each other with no gaps.
Comparing distributions using only center without mentioning spread.
A complete comparison always addresses both center (mean or median) AND spread (range, IQR, or MAD).
Mixed Review Practice
Guided Practice Video: Unit 9 Review
Watch this full Unit 9 review covering measures of center, spread, dot plots, histograms, box plots, and comparing distributions before tackling the mixed review problems below.
Video by Sang Real Math
Watch on YouTube ↗Find the mean, median, and mode: 14, 18, 14, 22, 30, 14, 26, 20, 18.
Find the range, IQR, and MAD for: 3, 7, 11, 15, 19.
Data: 10, 14, 18, 22, 26, 30, 34, 38, 42, 46. Find the five-number summary.
Data: 5, 8, 10, 12, 15, 18, 20, 22, 25, 80. Is 80 an outlier? Show your work using the IQR fence method.
A dot plot shows values: 2, 3, 3, 4, 4, 4, 5, 5, 6, 7. Find the median and IQR. Describe the shape.
A histogram peaks in the 40–49 range and is roughly symmetric. Is the mean likely greater than, less than, or approximately equal to the median? Explain.
Box plot: Min=12, Q1=20, Median=28, Q3=38, Max=50. Find the IQR and range. Describe the shape.
Data: 6, 9, 12, 15, 18, 21, 24. Find the MAD.
A data set has Mean=45 and Median=38. Is it right-skewed or left-skewed? Which measure of center is more appropriate?
Two box plots: Set A has Median=60, IQR=20. Set B has Median=60, IQR=5. Which is more consistent? Explain.
Data: 100, 105, 110, 115, 120, 125, 130, 135, 140, 200. Find the five-number summary. Is 200 an outlier?
A dot plot for Team A shows values clustered tightly around 8. A dot plot for Team B shows values spread from 2 to 14. Both have median=8. Which team is more consistent?
Histogram: 0–9: 1, 10–19: 3, 20–29: 8, 30–39: 10, 40–49: 6, 50–59: 2. Describe the shape. Estimate the median interval.
Box plot: Min=5, Q1=10, Median=12, Q3=18, Max=60. Describe the shape. Is the mean likely higher or lower than the median?
Data: 20, 22, 24, 26, 28, 30, 32, 34, 36, 38. Find the mean, median, and MAD. What type of distribution is this?
Two data sets: Set P has Mean=50, Median=49, IQR=12. Set Q has Mean=50, Median=38, IQR=10. Explain why these sets are NOT equally distributed.
A data set has Q1=30, Q3=50. What is the IQR? What are the outlier fences? A value of 75 — is it an outlier?
Compare: Group A has Median=72, IQR=18, right-skewed. Group B has Median=85, IQR=8, symmetric. Write a 2–3 sentence comparison.
Data: 15, 15, 15, 20, 25, 30, 35, 35, 35. Find the mean, median, and mode. Which measure of center best represents this data?
A box plot has a very long right whisker and the median is close to Q1. Describe the shape. Is the mean greater than or less than the median?
Two histograms: Histogram A peaks in 60–69 (symmetric). Histogram B peaks in 80–89 (left-skewed). Which class has a higher center? Which measure of center is better for Histogram B?
Data: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40. Find the five-number summary and IQR. Are there any outliers?
A student says "Set A is better because its mean is higher." Set A: 5, 5, 5, 5, 80. Set B: 20, 22, 24, 26, 28. Calculate mean and median for each. Is the student correct?
Explain the difference between range and IQR. When is IQR a better measure of spread than range?
A data set has Median=40, Q1=32, Q3=48. Find the IQR. Calculate the outlier fences. A value of 20 — is it an outlier?
Challenge Problems
- A data set of 10 values has a median of 25, Q1=18, Q3=32, and one outlier at 60. Without knowing all the values, what can you determine about the mean? Will it be greater than or less than 25? Explain your reasoning.
- Two data sets each have 12 values. Set A has a mean of 50 and MAD of 8. Set B has a mean of 50 and MAD of 2. Describe what each data set likely looks like. Which would you prefer if you wanted consistent results? Construct a possible example for each set.
- A company reports that the "average" salary is $75,000. The data: $30K, $32K, $35K, $38K, $40K, $42K, $45K, $48K, $50K, $240K. Calculate both the mean and median. Which is more representative? Why might the company prefer to report the mean? Why might employees prefer the median?
- Design two data sets of 8 values each where: (a) both have the same five-number summary, but (b) different means. Show all calculations to verify your design. Explain how this is possible.
- A researcher compares test scores from two teaching methods. Method A: Median=78, IQR=20, Range=55, right-skewed. Method B: Median=82, IQR=6, Range=30, symmetric. The researcher claims Method B is clearly superior. Write a complete statistical analysis (4–5 sentences) evaluating this claim. Include specific numbers and explain what each statistic means in context.
Unit 9 Summary
Ch 01 — Measures of Center: Mean (average), Median (middle), Mode (most frequent). Use median for skewed data or outliers.
Ch 02 — Measures of Spread: Range (Max−Min), IQR (Q3−Q1), MAD (avg distance from mean). IQR is resistant to outliers.
Ch 03 — Dot Plots & Histograms: Visualize individual values (dot plot) or intervals (histogram). Describe shape: symmetric, skewed, uniform, bimodal.
Ch 04 — Box Plots: Five-number summary (Min, Q1, Median, Q3, Max). Outlier fences: Q1−1.5(IQR) and Q3+1.5(IQR).
Ch 05 — Comparing Distributions: Compare center, spread, shape, and outliers. Always cite specific numbers. Write evidence-based conclusions.
Key Principle: Same mean ≠ same distribution. Always analyze center, spread, shape, AND outliers together for a complete picture.