📊 Stats & Data — Math Workbook

§1

Measures of Center

CORE

MEAN = add 'em all ÷ count | MEDIAN = middle value (sort first!) | MODE = most frequent
⚠️ Trap: outlier pulls the mean but NOT the median!

📝 Quick Example

Data: 4, 7, 7, 9, 13 → Mean = (4+7+7+9+13)÷5 = 8 | Median = 7 (middle) | Mode = 7

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Q1. A student scored $72, 85, 85, 90, 68$ on five tests. What is the mean score?

💡 Explanation

Add all scores: 72 + 85 + 85 + 90 + 68 = 400
Divide by 5: 400 ÷ 5 = 80 ✓
Watch out! 85 is the MODE (appears twice), not the mean.

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Q2. The ages of 5 kids are $8, 10, 10, 11, 6$ . Which value is the median?

💡 Explanation

First SORT: 6, 8, 10, 10, 11
With 5 values, the middle is the 3rd value → 10 ✓
⚠️ Always sort BEFORE finding the median!

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Q3. A class has test scores: $70, 72, 74, 75, 76, 100$ . A new student joins with score 100. Which measure of center changes the most?

💡 Explanation

100 is an outlier (far from the rest).
The mean is pulled strongly by outliers.
The median barely moves — it's resistant to outliers! ✓

§2

Range & Interquartile Range (IQR)

TRICKY

RANGE = Max − Min (measures total spread)
IQR = Q3 − Q1 (middle 50% spread — ignores outliers!)
Q1 = median of lower half | Q3 = median of upper half
⚠️ Trap: do NOT include the median when finding Q1 & Q3 (odd data set)

📝 Quick Example

Data (sorted): 2, 5, 7, 9, 12, 15, 18
Median = 9 → Lower half: 2, 5, 7 → Q1 = 5
Upper half: 12, 15, 18 → Q3 = 15
IQR = 15 − 5 = 10 | Range = 18 − 2 = 16

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Q4. The heights (cm) of 7 students are: $142, 155, 158, 163, 167, 171, 180$ . What is the range?

💡 Explanation

Range = Max − Min = 180 − 142 = 38 cm ✓

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Q5. Using the same data: $142, 155, 158, 163, 167, 171, 180$ , what is the IQR?

💡 Explanation

Median = 163 (4th value)
Lower half: 142, 155, 158 → Q1 = 155
Upper half: 167, 171, 180 → Q3 = 171
IQR = 171 − 155 = 16 ✓

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Q6. A teacher says "I want to describe the spread of the middle half of students' scores, ignoring the very high and very low scores." Which measure should she use?

💡 Explanation

The IQR describes the middle 50% of data.
It is resistant to outliers — perfect for ignoring extremes! ✓
Range includes all values (affected by outliers).

§3

Box Plots (Box-and-Whisker)

TRICKY

A box plot shows: Min → Q1 → Median → Q3 → Max
Box = Q1 to Q3 (IQR) | Whiskers = to min & max
⚠️ Trap: the box does NOT have to be centered — skewed data makes it lopsided!

📝 Box Plot Visual

Min
10 Q1
25 Med
35 Q3
50 Max
70

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Q7. From the box plot above (Min=10, Q1=25, Median=35, Q3=50, Max=70), what is the IQR?

💡 Explanation

IQR = Q3 − Q1 = 50 − 25 = 25 ✓
The box itself visually represents the IQR!

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Q8. A box plot has: Min=5, Q1=10, Median=12, Q3=20, Max=45. Which statement is BEST supported?

💡 Explanation

Max=45 is FAR from Q3=20 — this is an outlier!
The long right whisker shows the data is skewed right.
The median (12) is closer to Q1 than Q3, confirming right skew. ✓

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Q9. Two box plots show quiz scores for Class A and Class B. Both have the same median (75). Class A's IQR = 8, Class B's IQR = 22. What can you conclude?

💡 Explanation

Same median → same center. But IQR measures spread.
Class A: IQR = 8 → scores are clustered closely together.
Class B: IQR = 22 → scores are more spread out. ✓
Larger IQR = more variability!

§4

Stem-and-Leaf & Dot Plots

CORE

STEM = tens digit | LEAF = ones digit
To find median from stem-and-leaf: count from both ends inward
DOT PLOT = each dot = one data value above a number line

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Q10. Use this stem-and-leaf plot:

4

2 5 8 5

0 3 3 7 6

1 4

Key: 4|2 = 42

What is the median?

💡 Explanation

List all values: 42, 45, 48, 50, 53, 53, 57, 61, 64 → 9 values
Median = 5th value = 53 ✓

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Q11. The dot plot below shows the number of books read by 8 students:

1

2

3

4

5

What is the mode?

💡 Explanation

Mode = most frequent value = tallest stack of dots.
The value 3 has 3 dots — the most! ✓

§5

Measures of Variability

HARD

VARIABILITY = how spread out data is
RANGE & IQR both measure variability (spread)
MEAN & MEDIAN measure center — NOT variability!
⚠️ Trap: "How close together" = variability (small = close, large = spread)

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Q12. Which question could be answered using a measure of variability?

💡 Explanation

Variability answers "how spread out?" questions.
"Difference between tallest and shortest" → Range → variability ✓
"Average" → mean (center). "Most common" → mode (center).

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Q13. A coach wants to know how close together all runners' 100m times were. Which measure should the coach use?

💡 Explanation

"How close together" = how small the spread is → use a measure of variability.
Both Range and IQR work, but IQR is better as it ignores outliers.
✓ IQR tells us how bunched the middle half of runners are.

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Q14. Dataset A: $10, 10, 10, 10, 10$ | Dataset B: $2, 6, 10, 14, 18$ . Compare their ranges.

💡 Explanation

Dataset A: Range = 10 − 10 = 0 (no spread — all same!)
Dataset B: Range = 18 − 2 = 16
A has ZERO variability; B has much more spread. ✓

§6

Choosing the Right Measure

TRICKY

Use MEAN when: data is symmetric, no outliers
Use MEDIAN when: data has outliers or is skewed
Use MODE when: data is categorical (colors, names, etc.)
⚠️ Trap: real estate prices → use MEDIAN (billionaires skew mean up!)

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Q15. 5 people's salaries: $$30K, $32K, $34K, $35K, $200K$ . A journalist wants to report the "typical" salary. Which is BEST?

💡 Explanation

Mean = (30+32+34+35+200)÷5 = $66.2K — pulled up by the outlier!
Median = middle value = $34K — much more "typical" ✓
With outliers, MEDIAN represents "typical" better.

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Q16. Survey: "What is your favorite sport?" Results: Soccer×8, Basketball×5, Tennis×3, Swimming×2. Which measure of center is appropriate?

💡 Explanation

This is categorical data (sport names — not numbers!).
You can't find a mean or median of categories.
Use MODE → most frequent = Soccer ✓

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Q17. Adding a very large outlier to a dataset will:

💡 Explanation

A large outlier increases the mean significantly.
The median barely changes (shifts by 1 position at most).
The range increases (new max is the outlier).
Key: Mean is sensitive; Median is resistant. ✓

§7

Mixed Practice — Tricky Questions!

HARDEST

📊 5-Number Summary: Min, Q1, Median, Q3, Max
Each "section" of a box plot contains 25% of data.
⚠️ Even if a section LOOKS wider, it still holds 25% of the data!

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Q18. A box plot has Min=10, Q1=15, Median=20, Q3=30, Max=60. What percent of data falls between Q1 and Q3?

💡 Explanation

Q1 to Q3 = the BOX = the middle 50% of all data ✓
Q1 to Median = 25% | Median to Q3 = 25%
Total: 25% + 25% = 50%

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Q19. Dataset: $3, 5, 7, 9, 11, 13$ (6 values). What is the median?

💡 Explanation

6 values (even number) → median = average of 3rd and 4th values.
3rd value = 7, 4th value = 9
Median = (7 + 9) ÷ 2 = 8 ✓
⚠️ Even number of values: AVERAGE the two middle values!

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Q20. The mean of 4 numbers is 15. Three of the numbers are 10, 14, and 18. What is the 4th number?

💡 Explanation

Mean = Sum ÷ Count → Sum = Mean × Count = 15 × 4 = 60
Known sum: 10 + 14 + 18 = 42
4th number = 60 − 42 = 18 ✓

Statistics & DataAnalysis

Statistics & Data
Analysis