Statistics 11

AP / Honors · Data Analysis · Probability Foundations

20 Exam-Ready Problems · Self-Study Edition
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Mean & Median Box Plot (5-Number Summary) Standard Deviation Sample vs Population Normal Distribution Standard Normal (z-score) IQR & Outliers Empirical Rule (68-95-99.7) Word Problems

§1 — Mean & Median

Foundational
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⚡ Instant Memory Key
MEAN = SUM ÷ COUNT  |   MEDIAN = MIDDLE
Skewed right → Mean > Median  |  Skewed left → Mean < Median  |  Outlier pulls the MEAN, not the median.
Q 01 Easy
A student scored the following points on five quizzes:
Quiz Scores
82, 90, 78, 95, 75
What is the mean score?
Q 02 Easy
Find the median of the following data set:
Data
14, 7, 22, 3, 18, 11
Q 03 Word Problem
A real estate agent records the selling prices (in $1,000s) of 7 homes sold last month:
Home Prices ($1,000s)
210, 245, 198, 310, 225, 240, 890
Which measure of center best describes the typical home price, and why?
Box Plot · 5-Number Summary · IQR

§2 — Box Plot & IQR

Core
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MIN · Q1 · MEDIAN · Q3 · MAX
IQR = Q3 − Q1  |  Outlier if < Q1 − 1.5·IQR  OR  > Q3 + 1.5·IQR
Box = middle 50% of data  |  Whiskers = spread to min/max (excluding outliers)
Box Plot Anatomy
MIN Q1 Q2 Q3 MAX outlier ← IQR = Q3 − Q1 →
Q 04 Easy
Given the data set:  4, 8, 12, 16, 20, 24, 28

What is the IQR (Interquartile Range)?
Q 05 Medium
A box plot shows: Min = 10, Q1 = 25, Median = 40, Q3 = 55, Max = 70.

Is the value 85 considered an outlier?
Q 06 Word Problem
The ages (in years) of employees at two companies are summarized by box plots:

Company A: Min=22, Q1=28, Median=35, Q3=45, Max=60
Company B: Min=24, Q1=30, Median=38, Q3=42, Max=55

Which company has greater variability in employee ages?
Standard Deviation · Variance

§3 — Standard Deviation

Core
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SPREAD · DEVIATION · SQUARE · AVERAGE · ROOT
Steps: ① Find mean → ② Subtract mean from each value → ③ Square each → ④ Average the squares → ⑤ Take square root
Population: divide by n  |  Sample: divide by n−1  (Bessel's correction)
Key Formulas
Population SD: \(\sigma = \sqrt{\dfrac{\sum(x_i - \mu)^2}{N}}\)     Sample SD: \(s = \sqrt{\dfrac{\sum(x_i - \bar{x})^2}{n-1}}\)
Q 07 Medium
Which data set has a larger standard deviation?
Set A
10, 10, 10, 10, 10
Set B
2, 5, 10, 15, 18
Q 08 Word Problem
A quality control engineer measures the diameter (in mm) of 5 bolts from a sample:
Sample Data
10.1, 10.3, 10.0, 9.9, 10.2
The engineer calculates the mean to be 10.1 mm.

What is the sample variance? (Round to 4 decimal places)
Sample vs Population

§4 — Sample vs Population

Core
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PARAMETER (Population) · STATISTIC (Sample)
Greek letters = population: μ (mu), σ (sigma), N
Roman letters = sample: x̄ (x-bar), s, n
Sample SD divides by (n−1) to avoid underestimating population spread
Q 09 Medium
A researcher wants to know the average sleep duration of all American teenagers. She surveys 500 randomly selected teenagers.

In this context, which of the following is a parameter?
Q 10 ⚠ Tricky
You compute the standard deviation of test scores for a class of 30 students. You want to use this value to estimate the standard deviation for the entire school of 1,200 students.

Which formula should you use to calculate the class's standard deviation, and why?
Normal Distribution · Empirical Rule

§5 — Normal Distribution

Core
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⚡ Instant Memory Key — Empirical Rule
68 · 95 · 99.7
Within 1σ of mean → 68% of data
Within 2σ of mean → 95% of data
Within 3σ of mean → 99.7% of data
Bell curve: symmetric, mean = median = mode
Empirical Rule (68–95–99.7)
μ−2σ μ−σ μ μ+σ μ+2σ 68% 95% 99.7%
Q 11 Easy
IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.

According to the Empirical Rule, approximately what percentage of people have an IQ between 85 and 115?
Q 12 Word Problem
The birth weights of newborns at a hospital are normally distributed with a mean of 3,400 g and standard deviation of 500 g.

What percentage of newborns weigh more than 4,400 g?
Q 13 Hard
Heights of adult males are normally distributed: μ = 70 inches, σ = 3 inches.

What percentage of men are taller than 73 inches OR shorter than 64 inches?
z-Score · Standard Normal Distribution

§6 — z-Score & Standard Normal

Advanced
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z = (x − μ) ÷ σ
"How many standard deviations is x away from the mean?"
Standard normal: μ = 0, σ = 1  |  z > 0 → above mean  |  z < 0 → below mean
z-table gives P(Z < z) = area to the LEFT of z
z-Score Formula
\( z = \dfrac{x - \mu}{\sigma} \)     Standard Normal: \( Z \sim N(0, 1) \)
Q 14 Easy
A student scores 88 on an exam. The class mean is 80 and the standard deviation is 10.

What is the student's z-score?
Q 15 Word Problem
SAT Math scores are normally distributed with μ = 520 and σ = 115.

Alex scored 750. Maria scored 680 on a different exam with μ = 600 and σ = 80.

Who performed better relative to their peer group?
Q 16 Hard · Word Problem
Battery lifetimes are normally distributed with μ = 200 hours and σ = 25 hours.

Using the z-table: P(Z < 1.80) = 0.9641

What percentage of batteries last between 200 and 245 hours?
Q 17 Hard
For a standard normal distribution, P(Z < −1.5) = 0.0668.

What is P(−1.5 < Z < 1.5)?
Q 18 Hard · Word Problem
A coffee machine dispenses amounts that are normally distributed with μ = 8.0 oz and σ = 0.4 oz.

A cup overflows if it receives more than 8.8 oz. Cups overflow approximately what percentage of the time?
Q 19 ⭐ Expert
A college admission office finds that GPA scores are normally distributed with μ = 3.2 and σ = 0.4.

They want to admit only students scoring in the top 16%. What is the minimum GPA required for admission?
Q 20 ⭐ Expert · Word Problem
A teacher gives a statistics test. The results are approximately normally distributed.

Class Results
Mean = 72  |  Standard Deviation = 8  |  n = 30
The teacher decides to curve: all students more than 1.5 standard deviations below the mean will retake the test.

What is the cutoff score for retaking, and approximately how many students must retake?