Module 2 Practice Problems

Solution:

Step 1: Add all values
18 + 22 + 25 + 28 + 32 + 35 = 160

Step 2: Count how many values
There are 6 ages

Step 3: Divide sum by count
Mean = 160 ÷ 6 = 26.67 years

Answer: 26.67 years

Solution:

Step 1: Data is already ordered: 68, 72, 75, 80, 85, 90

Step 2: Count values: 6 (even number)

Step 3: Find the two middle values
With 6 values, the 3rd and 4th values are in the middle: 75 and 80

Step 4: Average them
Median = (75 + 80) ÷ 2 = 77.5

Answer: 77.5

Solution:

Count frequencies:

• 7 appears 1 time
• 8 appears 2 times
• 9 appears 3 times ← Most frequent!
• 10 appears 2 times
• 11 appears 1 time

Answer: Mode = 9 (appears most frequently)

Solution:

Mean:
Sum = 45 + 48 + 50 + 52 + 55 + 250 = 500
Mean = 500 ÷ 6 = $83.33k

Median:
Ordered: 45, 48, 50, 52, 55, 250
6 values (even), so average the 3rd and 4th values
Median = (50 + 52) ÷ 2 = $51k

Which is better?
Median ($51k) better represents a typical salary. Five out of six employees earn $45-55k. The $250k outlier (possibly CEO or owner) pulls the mean up to $83.33k, which doesn't represent most employees.

Answer: Mean = $83.33k, Median = $51k; Median is more representative

Solution:

a) Mean for average height
Why: Heights are typically symmetric with no extreme outliers. Mean uses all data and is appropriate.

b) Median for home prices
Why: Home prices are right-skewed with outliers (mansions). Median is resistant to these high outliers and better represents a "typical" home.

c) Mode for pizza toppings
Why: This is categorical data (pepperoni, mushroom, etc.). You can't calculate mean or median of text! Mode shows the most popular choice.

Solution:

Range:
Max = 15, Min = 6
Range = 15 - 6 = 9

IQR:
Step 1: Find median (Q2) = 10 (5th value of 9)
Step 2: Lower half: 6, 7, 8, 9 → Q1 = (7+8)/2 = 7.5
Step 3: Upper half: 11, 12, 14, 15 → Q3 = (12+14)/2 = 13
Step 4: IQR = Q3 - Q1 = 13 - 7.5 = 5.5

Answer: Range = 9, IQR = 5.5

Solution:

Step 1: Calculate mean
x̄ = (2+4+6+8+10) / 5 = 30/5 = 6

Step 2: Create deviation table

x	x - x̄	(x - x̄)²
2	2-6 = -4	16
4	4-6 = -2	4
6	6-6 = 0	0
8	8-6 = 2	4
10	10-6 = 4	16

Step 3: Sum squared deviations
Σ(x - x̄)² = 16 + 4 + 0 + 4 + 16 = 40

Step 4: Divide by (n-1)
s² = 40 / (5-1) = 40/4 = 10

Answer: Variance = 10

Solution:

Calculation:
Standard Deviation = √(Variance) = √25 = 5

Interpretation:
The standard deviation of 5 means that values in the dataset typically vary by about 5 units from the mean. This is in the original units (not squared like variance), making it more interpretable.

Answer: SD = 5

Solution:

a) Between 62 and 68 inches:
62 = 65 - 3 (mean - 1 SD)
68 = 65 + 3 (mean + 1 SD)
This is within ±1 SD → ~68% of people

b) Between 59 and 71 inches:
59 = 65 - 6 (mean - 2 SD)
71 = 65 + 6 (mean + 2 SD)
This is within ±2 SD → ~95% of people

Answers: a) 68%, b) 95%

Solution:

Class B is more consistent.

Reasoning:
Both classes have the same mean (75), but Class B has a much smaller standard deviation (4 vs. 12). A smaller SD means scores are clustered more tightly around the mean.

What this means:

• Class B (SD=4): Most scores are within 71-79 (very consistent performance)
• Class A (SD=12): Scores vary widely, from ~63 to ~87 (inconsistent performance)

Answer: Class B is more consistent (smaller SD = less variability)

Solution:

Analysis:
Mean (50) < Median (55) < Mode (60)

Shape: Left-skewed

Explanation:
When Mean < Median < Mode, the distribution has a long tail extending to the left (low values). The mean is pulled down by low outliers, while the mode (most common value) is on the right side.

Answer: Left-skewed distribution

Solution:

a) Right-skewed
Mean ($85k) > Median ($58k) indicates a right-skewed distribution.

b) Median ($58,000) is more representative
The median better represents a typical household because it's resistant to high-income outliers (millionaires, billionaires) that pull the mean up.

c) Why income is right-skewed:

• Most people earn moderate incomes ($40k-$80k)
• A small percentage earn very high incomes ($500k+)
• Income has a natural floor ($0) but no ceiling
• This creates a long right tail of high earners

This is why news reports use "median household income" rather than mean!

Solution:

Conclusion:
In a bell-shaped (symmetric) distribution:
Mean ≈ Median ≈ Mode

Therefore:
Median ≈ 100
Mode ≈ 100

Reasoning:
Bell-shaped distributions are symmetric, meaning they're balanced on both sides of the center. All three measures of center converge at the same point.

Answer: Median and Mode both ≈ 100

Solution:

Without outlier (Original):
Mean = (15+18+20+22+25+28+30)/7 = 158/7 = 22.57 minutes
Median = 22 (middle of 7 values)

With outlier (120 added):
Mean = (15+18+20+22+25+28+30+120)/8 = 278/8 = 34.75 minutes
Median = (22+25)/2 = 23.5 minutes

Changes:

• Mean: 22.57 → 34.75 (+12.18 minutes, 54% increase!)
• Median: 22 → 23.5 (+1.5 minutes, 7% increase)

Answer: The mean changed much more dramatically (54% vs 7%). This demonstrates that the median is resistant to outliers while the mean is sensitive to outliers.

Solution:

A) Kindergarten ages: Left-skewed
Most kids are 5-6 years old, with a few younger kids who started early creating a left tail.

B) Home prices: Right-skewed
Most homes are in an affordable range ($200k-$500k), but a few expensive mansions ($2M+) create a long right tail.

C) Heights of adult men: Symmetric
Heights follow a bell curve with most men near average (5'9"), fewer very short or very tall, creating a symmetric distribution.

Solution:

Data (already ordered): 12, 15, 18, 20, 22, 25, 28, 30, 35

Step 1: Min and Max
Min = 12, Max = 35

Step 2: Median (Q2)
9 values, so median is the 5th value = 22

Step 3: Q1 (median of lower half)
Lower half: 12, 15, 18, 20
Q1 = (15 + 18)/2 = 16.5

Step 4: Q3 (median of upper half)
Upper half: 25, 28, 30, 35
Q3 = (28 + 30)/2 = 29

Five-Number Summary:
Min = 12, Q1 = 16.5, Median = 22, Q3 = 29, Max = 35

Solution:

From Problem 16: Q1 = 16.5, Q3 = 29

Step 1: Calculate IQR
IQR = Q3 - Q1 = 29 - 16.5 = 12.5

Step 2: Calculate fences
Lower Fence = Q1 - (1.5 × IQR) = 16.5 - (1.5 × 12.5) = 16.5 - 18.75 = -2.25
Upper Fence = Q3 + (1.5 × IQR) = 29 + (1.5 × 12.5) = 29 + 18.75 = 47.75

Step 3: Check for outliers
Data: 12, 15, 18, 20, 22, 25, 28, 30, 35
Is any value < -2.25? No
Is any value > 47.75? No

Answer: No outliers (all values fall within the fences)

Solution:

a) IQR:
IQR = Q3 - Q1 = 80 - 40 = 40

b) Percentage between 40 and 80:
Q1 (40) is the 25th percentile, Q3 (80) is the 75th percentile.
75% - 25% = 50% of data falls between Q1 and Q3

c) Shape analysis:

• Median (45) is closer to Q1 (40) than to Q3 (80)
• Distance Q1 to Median = 5
• Distance Median to Q3 = 35 (much larger!)
• Upper whisker (80 to 120) is longer than lower whisker (20 to 40)

Shape: Right-skewed (median near Q1, longer upper whisker)

Answers: a) IQR = 40, b) 50%, c) Right-skewed

Solution:

a) Higher typical score: Class A
Class A median = 80 > Class B median = 75
The median represents the typical score, so Class A performed better.

b) More consistent scores: Class A
Class A IQR = 88 - 70 = 18
Class B IQR = 85 - 65 = 20
Smaller IQR = more consistent (middle 50% more tightly clustered)

c) More variability overall: Class B
Class A Range = 95 - 60 = 35
Class B Range = 100 - 50 = 50
Class B has a wider range (more spread from lowest to highest)

Summary: Class A performed better on average with more consistent scores, while Class B had more variability (both higher highs and lower lows).

Solution:

a) Five-Number Summary:

Min = 5, Max = 95
Median (6th of 11 values) = 22
Q1 (median of 5, 8, 12, 15, 18) = 12
Q3 (median of 25, 30, 35, 40, 95) = 35

Five-number summary: 5, 12, 22, 35, 95

b) Outlier Detection:
IQR = 35 - 12 = 23
Lower Fence = 12 - (1.5 × 23) = 12 - 34.5 = -22.5
Upper Fence = 35 + (1.5 × 23) = 35 + 34.5 = 69.5

Is 95 > 69.5? YES! 95 is an outlier

c) Boxplot Description:

• Box: From Q1 (12) to Q3 (35)
• Median line: At 22, slightly right of center in the box
• Lower whisker: From 12 down to 5 (length = 7)
• Upper whisker: From 35 to 40 (NOT to 95! Stops at last non-outlier)
• Outlier point: A dot at 95, beyond the upper whisker
• Shape: Appears slightly right-skewed due to the outlier

Answers: a) (5, 12, 22, 35, 95), b) Yes, 95 is an outlier, c) Box from 12-35 with median at 22, upper whisker to 40, outlier dot at 95