§1 — Mean & Median
⚡ Quick Memory Points
MEAN = BALANCE POINT
Add all values ÷ count. Pulled by outliers.
MEDIAN = MIDDLE VALUE
Sort first. Resistant to outliers.
SKEW TRICK
Right skew → mean > median. Left skew → mean < median.
ODD vs EVEN
Odd n: middle value. Even n: average of two middle values.
01
Finding the Mean from a Data Set
A student recorded their quiz scores for the week:
72, 85, 90, 68, 95
What is the mean score?
72, 85, 90, 68, 95
What is the mean score?
✓ Correct Answer: B — 82
Mean = (72 + 85 + 90 + 68 + 95) ÷ 5 = 410 ÷ 5 = 82
Mean = (72 + 85 + 90 + 68 + 95) ÷ 5 = 410 ÷ 5 = 82
Mean (x̄) = Σx / n = (sum of all values) / (number of values)
Always add first, then divide by how many values there are.
02
Finding the Median — Even Number of Values
Six students reported how many hours they slept last night:
7, 5, 9, 6, 8, 4
What is the median number of hours?
7, 5, 9, 6, 8, 4
What is the median number of hours?
💡 Sort the data first: 4, 5, 6, 7, 8, 9 — then find the middle.
✓ Correct Answer: C — 6.5
Sorted: 4, 5, 6, 7, 8, 9
With an even number of values (n = 6), take the average of the 3rd and 4th values:
Median = (6 + 7) / 2 = 6.5
Sorted: 4, 5, 6, 7, 8, 9
With an even number of values (n = 6), take the average of the 3rd and 4th values:
Median = (6 + 7) / 2 = 6.5
Even n: Median = (n/2 th value + (n/2 + 1)th value) / 2
03
Outlier Effect — Mean vs. Median
The annual salaries (in thousands of dollars) at a small company are:
42, 45, 47, 48, 51, 210
The CEO earns $210,000. Which statement is best?
42, 45, 47, 48, 51, 210
The CEO earns $210,000. Which statement is best?
✓ Correct Answer: D
Mean = (42+45+47+48+51+210)/6 = 443/6 ≈ 73.8 — pulled way up by the CEO's salary.
Sorted: 42, 45, 47, 48, 51, 210 → Median = (47+48)/2 = 47.5
The median of $47,500 is much closer to what most employees actually earn. When outliers are present, median is the preferred measure of center.
Mean = (42+45+47+48+51+210)/6 = 443/6 ≈ 73.8 — pulled way up by the CEO's salary.
Sorted: 42, 45, 47, 48, 51, 210 → Median = (47+48)/2 = 47.5
The median of $47,500 is much closer to what most employees actually earn. When outliers are present, median is the preferred measure of center.
§2 — Box Plot & IQR
⚡ Quick Memory Points
5-NUMBER SUMMARY
Min · Q1 · Median · Q3 · Max
IQR = Q3 − Q1
The "middle 50%" spread. Resistant to outliers.
OUTLIER FENCE
Below Q1 − 1.5×IQR or above Q3 + 1.5×IQR
BOX WIDTH = SPREAD
Wider box = more spread in the middle 50%
04
Calculating IQR from a Five-Number Summary
A data set has the following five-number summary:
What is the Interquartile Range (IQR)?
| Min | Q1 | Median | Q3 | Max |
|---|---|---|---|---|
| 12 | 20 | 28 | 35 | 50 |
✓ Correct Answer: B — 15
IQR = Q3 − Q1 = 35 − 20 = 15
The IQR represents the range of the middle 50% of the data. It is NOT the full range (Max − Min = 38).
05
Identifying Outliers Using the IQR Rule
A data set has Q1 = 18 and Q3 = 30. Using the 1.5 × IQR rule, which value would be considered an outlier?
Outlier if: value < Q1 − 1.5·IQR OR value > Q3 + 1.5·IQR
✓ Correct Answer: C — 52
IQR = Q3 − Q1 = 30 − 18 = 12
Lower fence = 18 − 1.5(12) = 18 − 18 = 0
Upper fence = 30 + 1.5(12) = 30 + 18 = 48
Any value < 0 or > 48 is an outlier. Only 52 > 48, so 52 is the outlier.
Note: 5 and 14 are above 0, so they are NOT outliers. Always compute the fences first!
IQR = Q3 − Q1 = 30 − 18 = 12
Lower fence = 18 − 1.5(12) = 18 − 18 = 0
Upper fence = 30 + 1.5(12) = 30 + 18 = 48
Any value < 0 or > 48 is an outlier. Only 52 > 48, so 52 is the outlier.
Note: 5 and 14 are above 0, so they are NOT outliers. Always compute the fences first!
06
Reading a Box Plot — What Percentage Lies Between Q1 and Q3?
The box plot below represents test scores for a class of 40 students:
Approximately what percentage of students scored between 58 and 84?
✓ Correct Answer: B — 50%
The box in a box plot spans from Q1 to Q3, which always contains the middle 50% of the data.
• Below Q1 (left whisker) → 25%
• Q1 to Median → 25%
• Median to Q3 → 25%
• Above Q3 (right whisker) → 25%
Q1 to Q3 = 25% + 25% = 50%
The box in a box plot spans from Q1 to Q3, which always contains the middle 50% of the data.
• Below Q1 (left whisker) → 25%
• Q1 to Median → 25%
• Median to Q3 → 25%
• Above Q3 (right whisker) → 25%
Q1 to Q3 = 25% + 25% = 50%
§3 — Standard Deviation & Variance
⚡ Quick Memory Points
SD = AVERAGE DISTANCE
How far data points are from the mean, on average.
POPULATION: σ, divide by N
σ² = Σ(x−μ)²/N
SAMPLE: s, divide by n−1
s² = Σ(x−x̄)²/(n−1) ← Bessel's correction
LARGER SD = MORE SPREAD
SD = 0 means all values are identical.
07
Interpreting Standard Deviation
Two classes took the same test. Class A has a mean of 75 with a standard deviation of 3. Class B has a mean of 75 with a standard deviation of 15.
Which statement is true?
Which statement is true?
✓ Correct Answer: A
Both classes have the same mean (75), so we cannot say one performed "better." The key difference is spread.
A larger standard deviation means scores are more spread around the mean. Class B (σ = 15) has scores that vary much more widely than Class A (σ = 3).
Both classes have the same mean (75), so we cannot say one performed "better." The key difference is spread.
A larger standard deviation means scores are more spread around the mean. Class B (σ = 15) has scores that vary much more widely than Class A (σ = 3).
08
Sample vs. Population — Which Formula?
A researcher wants to estimate the average height of all high school students in the United States. She measures the heights of 200 randomly selected students.
To calculate the standard deviation of her sample, she should divide by:
To calculate the standard deviation of her sample, she should divide by:
Population SD: σ = √[Σ(x − μ)² / N]
Sample SD: s = √[Σ(x − x̄)² / (n − 1)]
✓ Correct Answer: C — n − 1 = 199
When computing a sample standard deviation, we divide by n − 1 (called Bessel's correction). This makes the estimate slightly larger, correcting for the fact that a sample tends to underestimate the true population spread.
Key rule:
• Entire group studied → Population → divide by N
• Subset studied → Sample → divide by n − 1
When computing a sample standard deviation, we divide by n − 1 (called Bessel's correction). This makes the estimate slightly larger, correcting for the fact that a sample tends to underestimate the true population spread.
Key rule:
• Entire group studied → Population → divide by N
• Subset studied → Sample → divide by n − 1
09
Computing Population Standard Deviation
A population has five values: 2, 4, 4, 4, 6. The mean is 4.
What is the population standard deviation σ?
What is the population standard deviation σ?
σ = √[ Σ(xᵢ − μ)² / N ]
✓ Correct Answer: B — √2 ≈ 1.41
Deviations from mean (μ = 4):
(2−4)² = 4 | (4−4)² = 0 | (4−4)² = 0 | (4−4)² = 0 | (6−4)² = 4
Sum of squared deviations = 4 + 0 + 0 + 0 + 4 = 8
Variance σ² = 8 / 5 = 1.6
σ = √1.6 = √(8/5) = √2 ≈ 1.414
Deviations from mean (μ = 4):
(2−4)² = 4 | (4−4)² = 0 | (4−4)² = 0 | (4−4)² = 0 | (6−4)² = 4
Sum of squared deviations = 4 + 0 + 0 + 0 + 4 = 8
Variance σ² = 8 / 5 = 1.6
σ = √1.6 = √(8/5) = √2 ≈ 1.414
§4 — Normal Distribution
⚡ Quick Memory Points
68-95-99.7 RULE
±1σ: 68% | ±2σ: 95% | ±3σ: 99.7%
SYMMETRIC BELL
Mean = Median = Mode. Mirror image.
AREA = PROBABILITY
Area under the curve between two values = P(a < X < b)
TOTAL AREA = 1 (100%)
The entire curve represents all possible outcomes.
10
The 68-95-99.7 Empirical Rule
The heights of adult males in a city are normally distributed with a mean of μ = 70 inches and standard deviation σ = 3 inches.
What percentage of men have heights between 67 and 73 inches?
What percentage of men have heights between 67 and 73 inches?
✓ Correct Answer: B — 68%
67 = 70 − 3 = μ − 1σ
73 = 70 + 3 = μ + 1σ
The interval (μ − σ, μ + σ) always contains 68% of the data in a normal distribution.
67 = 70 − 3 = μ − 1σ
73 = 70 + 3 = μ + 1σ
The interval (μ − σ, μ + σ) always contains 68% of the data in a normal distribution.
Empirical Rule: μ ± 1σ → 68% | μ ± 2σ → 95% | μ ± 3σ → 99.7%
11
One-Sided Probability with the Empirical Rule
IQ scores are normally distributed with μ = 100 and σ = 15.
What percentage of people have an IQ greater than 130?
What percentage of people have an IQ greater than 130?
💡 130 = 100 + 2(15) = μ + 2σ. Use the 95% rule and symmetry.
✓ Correct Answer: C — 2.5%
Between μ − 2σ (70) and μ + 2σ (130) lies 95% of the data.
That leaves 5% outside this range. By symmetry, half is on each tail:
P(X > 130) = 5% / 2 = 2.5%
A common mistake: choosing 5% (forgetting to split between two tails).
Between μ − 2σ (70) and μ + 2σ (130) lies 95% of the data.
That leaves 5% outside this range. By symmetry, half is on each tail:
P(X > 130) = 5% / 2 = 2.5%
A common mistake: choosing 5% (forgetting to split between two tails).
§5 — z-Scores & Standard Normal Distribution
⚡ Quick Memory Points
z = (x − μ) / σ
How many SDs above/below the mean. Unit = "standard deviations."
STANDARD NORMAL
N(0,1) — mean=0, SD=1. Use z-table for probabilities.
NEGATIVE z = BELOW MEAN
z = −1.5 means 1.5 SDs below the mean.
z-TABLE GIVES P(Z < z)
For P(Z > z): use 1 − table value.
12
Computing a z-Score
A student scored 88 on an exam where the class mean was μ = 80 and the standard deviation was σ = 5.
What is the student's z-score?
What is the student's z-score?
z = (x − μ) / σ
✓ Correct Answer: C — 1.6
z = (88 − 80) / 5 = 8 / 5 = 1.6
This means the student scored 1.6 standard deviations above the mean. A positive z-score always means above average.
z = (88 − 80) / 5 = 8 / 5 = 1.6
This means the student scored 1.6 standard deviations above the mean. A positive z-score always means above average.
13
Using the z-Table — P(Z < z)
According to a standard normal z-table, P(Z < 1.25) = 0.8944.
What is P(Z > 1.25)?
What is P(Z > 1.25)?
💡 Total area under the standard normal curve = 1
✓ Correct Answer: B — 0.1056
P(Z > 1.25) = 1 − P(Z < 1.25) = 1 − 0.8944 = 0.1056
The z-table gives the area to the left. To find the area to the right, always subtract from 1.
P(Z > 1.25) = 1 − P(Z < 1.25) = 1 − 0.8944 = 0.1056
The z-table gives the area to the left. To find the area to the right, always subtract from 1.
14
Probability Between Two z-Values
Using the z-table:
P(Z < 1.50) = 0.9332 P(Z < −0.50) = 0.3085
What is P(−0.50 < Z < 1.50)?
P(Z < 1.50) = 0.9332 P(Z < −0.50) = 0.3085
What is P(−0.50 < Z < 1.50)?
P(a < Z < b) = P(Z < b) − P(Z < a)
✓ Correct Answer: A — 0.6247
P(−0.50 < Z < 1.50) = P(Z < 1.50) − P(Z < −0.50)
= 0.9332 − 0.3085 = 0.6247
Think of the curve as area. Subtract the left portion you don't want from the total left area up to the right boundary.
P(−0.50 < Z < 1.50) = P(Z < 1.50) − P(Z < −0.50)
= 0.9332 − 0.3085 = 0.6247
Think of the curve as area. Subtract the left portion you don't want from the total left area up to the right boundary.
§6 — Exam-Level Word Problems
15
Comparing Scores Across Different Tests
Maria scored 78 on a Math test (μ = 70, σ = 8) and 82 on an English test (μ = 78, σ = 6).
On which test did Maria perform relatively better compared to her classmates?
On which test did Maria perform relatively better compared to her classmates?
✓ Correct Answer: B — Math (higher z-score)
Math z = (78 − 70) / 8 = 8/8 = 1.00
English z = (82 − 78) / 6 = 4/6 ≈ 0.67
A higher z-score means she performed better relative to the class distribution. Her Math z = 1.00 > English z = 0.67, so she did relatively better in Math. Never compare raw scores across different distributions!
Math z = (78 − 70) / 8 = 8/8 = 1.00
English z = (82 − 78) / 6 = 4/6 ≈ 0.67
A higher z-score means she performed better relative to the class distribution. Her Math z = 1.00 > English z = 0.67, so she did relatively better in Math. Never compare raw scores across different distributions!
16
Quality Control — Bottling Plant
A bottling plant fills bottles with a mean of μ = 500 mL and standard deviation σ = 5 mL. The fill amounts follow a normal distribution. A bottle is rejected if it contains less than 492 mL.
Use the z-table: P(Z < −1.60) = 0.0548
Approximately what percentage of bottles will be rejected?
Use the z-table: P(Z < −1.60) = 0.0548
Approximately what percentage of bottles will be rejected?
✓ Correct Answer: B — 5.48%
Step 1: Find the z-score for 492 mL:
z = (492 − 500) / 5 = −8/5 = −1.60
Step 2: P(X < 492) = P(Z < −1.60) = 0.0548 = 5.48%
The z-table gives the probability to the left, which is exactly what we want (less than 492 mL).
Step 1: Find the z-score for 492 mL:
z = (492 − 500) / 5 = −8/5 = −1.60
Step 2: P(X < 492) = P(Z < −1.60) = 0.0548 = 5.48%
The z-table gives the probability to the left, which is exactly what we want (less than 492 mL).
17
Skewness — Which Measure to Report?
A city reports the distribution of household incomes. The data is strongly right-skewed due to a few very high-income households.
A journalist wants to report a single number that best represents the typical household income. Which should they use, and why?
A journalist wants to report a single number that best represents the typical household income. Which should they use, and why?
✓ Correct Answer: C — Median
In a right-skewed distribution, the mean is pulled toward the high tail (higher incomes). This makes the mean misleadingly high as a "typical" value.
The median is the preferred measure of center for skewed data because it is resistant to extreme values. This is why you often see "median household income" in news reports rather than "mean household income."
In a right-skewed distribution, the mean is pulled toward the high tail (higher incomes). This makes the mean misleadingly high as a "typical" value.
The median is the preferred measure of center for skewed data because it is resistant to extreme values. This is why you often see "median household income" in news reports rather than "mean household income."
18
Effect of Adding a Constant on Mean and SD
A class scored poorly on a test. The teacher decides to add 10 points to every student's score.
The original mean was 65 and the original standard deviation was 8.
After adding 10 points to every score, what are the new mean and new standard deviation?
The original mean was 65 and the original standard deviation was 8.
After adding 10 points to every score, what are the new mean and new standard deviation?
✓ Correct Answer: B — Mean = 75, SD = 8
When you add (or subtract) a constant to every value:
• The mean shifts by that constant → 65 + 10 = 75
• The standard deviation stays the same → still 8
Why? Adding a constant moves all points equally. The distances between points don't change, so spread is unchanged. Only the center shifts.
When you add (or subtract) a constant to every value:
• The mean shifts by that constant → 65 + 10 = 75
• The standard deviation stays the same → still 8
Why? Adding a constant moves all points equally. The distances between points don't change, so spread is unchanged. Only the center shifts.
19
Working Backwards — Finding a Raw Score from a Percentile
Exam scores are normally distributed with μ = 74 and σ = 10.
Using the z-table: P(Z < 1.28) = 0.8997 ≈ 90th percentile
A student wants to score in the top 10% of the class. What is the minimum score she needs?
Using the z-table: P(Z < 1.28) = 0.8997 ≈ 90th percentile
A student wants to score in the top 10% of the class. What is the minimum score she needs?
💡 Reverse formula: x = μ + z · σ
✓ Correct Answer: D — 86.8
Top 10% means P(X > x) = 0.10, so P(X < x) = 0.90. The z-value for 90th percentile ≈ 1.28.
x = μ + z · σ = 74 + (1.28)(10) = 74 + 12.8 = 86.8
This "un-standardizing" process: take the z-score and convert back to the original units using x = μ + zσ.
Top 10% means P(X > x) = 0.10, so P(X < x) = 0.90. The z-value for 90th percentile ≈ 1.28.
x = μ + z · σ = 74 + (1.28)(10) = 74 + 12.8 = 86.8
This "un-standardizing" process: take the z-score and convert back to the original units using x = μ + zσ.
20
The Coffee Shop Problem — Mean, SD, and z-Scores
A coffee shop tracks daily customer counts. The data is approximately normally distributed with μ = 240 customers/day and σ = 30 customers/day.
The shop needs at least 270 customers per day to be profitable, and the manager wants to know how often profit occurs.
Use: P(Z < 1.00) = 0.8413
On what percentage of days is the shop NOT profitable (fewer than 270 customers)?
The shop needs at least 270 customers per day to be profitable, and the manager wants to know how often profit occurs.
Use: P(Z < 1.00) = 0.8413
On what percentage of days is the shop NOT profitable (fewer than 270 customers)?
✓ Correct Answer: A — 84.13%
Step 1: Compute z for 270:
z = (270 − 240) / 30 = 30/30 = 1.00
Step 2: P(X < 270) = P(Z < 1.00) = 0.8413 = 84.13%
"Not profitable" means fewer than 270 customers, which is the left-tail area. The shop is profitable only about 15.87% of days (1 − 0.8413).
Tip: Always identify "less than" vs "greater than" before deciding to subtract from 1.
Step 1: Compute z for 270:
z = (270 − 240) / 30 = 30/30 = 1.00
Step 2: P(X < 270) = P(Z < 1.00) = 0.8413 = 84.13%
"Not profitable" means fewer than 270 customers, which is the left-tail area. The shop is profitable only about 15.87% of days (1 − 0.8413).
Tip: Always identify "less than" vs "greater than" before deciding to subtract from 1.
0/20
Complete all 20 questions to see your score.