Find the Center
Find the mean:
4, 6, 8, 10
\frac{4+6+8+10}{4}=\frac{28}{4}=7
Find the range:
12, 7, 15, 9, 11
8
A student is at the 80th percentile.
Does this mean the student answered 80% of the questions correctly? Explain.
No.
A percentile describes relative position in a group.
It does not give the percentage of questions answered correctly.
A survey records each student’s favorite music genre.
Are these data qualitative or quantitative?
Qualitative.
Music genres are categories rather than numerical measurements.
A researcher surveys 80 students at a school with 900 students.
Identify the sample and population.
Sample: the 80 surveyed students.
Population: all 900 students at the school.
Find the median and mode:
9, 3, 7, 3, 8, 6
Median = 6.5
Mode = 3
Both sets have a mean of 10.
Set A: 9, 10, 11
Set B: 0, 10, 20
Which has the greater standard deviation?
Explain.
Set B.
Its values lie farther from the mean, so its standard deviation is greater.
A test has a mean of 70 and a standard deviation of 8.
Find the z-score for a score of 86.
What does it mean?
z=\frac{86-70}{8}=\frac{16}{8}=2
The score is two standard deviations above the mean.
A frequency table contains:
0–4: 3 students
5–9: 7 students
10–14: 5 students
How many students are represented?
What is the modal class?
15 students.
Modal class: 5–9.
Total frequency: 3 + 7 + 5 = 15
The modal class has the greatest frequency.
A factory randomly selects a starting phone and then tests every 100th phone produced.
What sampling method is being used?
Systematic sampling.
After a random start, phones are selected at a fixed interval.
A student earns these scores:
Homework: 90, worth 20%
Quizzes: 80, worth 30%
Tests: 70, worth 50%
Find the weighted average.
90(0.20) + 80(0.30) + 70(0.50)
= 18 + 24 + 35
= 77
A data set has a variance of 49 minutes².
What is its standard deviation? Include units.
7 minutes
Find the five-number summary:
2, 4, 6, 8, 10, 12, 14, 16
Minimum = 2
Q₁ = 5
Median = 9
Q₃ = 13
Maximum = 16
A frequency distribution has classes:
10–19
20–29
30–39
40–49
What is the class width?
*Bonus* What are the boundaries of 20–29 if measurements were rounded to the nearest whole unit?
Class width = 10
20 − 10 = 10
Class boundaries: 19.5 and 29.5
Identify each sampling method:
A. Randomly select students from EVERY grade.
B. Randomly select three homerooms and survey EVERYONE in those rooms.
A: Stratified sampling.
A sample comes from every grade.
B: Cluster sampling.
Everyone in the selected groups is surveyed.
Five salaries are:
$30,000; $32,000; $34,000; $36,000; $168,000
Which better represents a typical salary: the mean or median? Explain.
The median: $34,000.
The unusually high salary pulls the mean up to $60,000.
The median is resistant to extreme values.
A bell-shaped distribution has a mean of 50 and a standard deviation of 5.
Approximately what percentage of values fall between 40 and 60?
95%
A distribution has Q₁ = 12 and Q₃ = 20.
Using the 1.5 × IQR rule, determine whether 33 is an outlier.
Yes.
IQR = 20 − 12 = 8
Lower fence: 12 − 1.5(8) = 0
Upper fence: 20 + 1.5(8) = 32
33 is above the upper fence.
A distribution has a long tail extending toward larger values.
Identify the skew direction.
Which would usually be greater: the mean or median?
Right-skewed, or positively skewed.
The mean would usually exceed the median.
Larger values pull the mean toward the right tail.
A school posts an optional online survey asking whether homework should be eliminated.
Can the responses safely represent all students?
Identify the main concern.
No. Voluntary response bias is a concern.
Students choose whether to respond.
Those with strong opinions may be more likely to participate, making the respondents unrepresentative.
Four test scores have a mean of 78.
After a fifth score is added, the mean becomes 82.
What is the fifth score?
98
The values 2, 4, and 6 are a SAMPLE.
Find the sample variance and sample standard deviation.
Sample variance: s^2=4
Sample standard deviation: s=2
Mean = 4
Squared deviations: 4, 0, 4
s^2=\frac{4+0+4}{3-1}=\frac{8}{2}=4
s=\sqrt{4}=2
Two boxplots have these five-number summaries:
A: 2, 6, 10, 14, 18
B: 2, 9, 10, 11, 18
Compare their medians, ranges, and IQRs.
Which has a more tightly clustered middle 50%?
Both medians = 10.
Both ranges = 18 − 2 = 16.
A: IQR = 14 − 6 = 8
B: IQR = 11 − 9 = 2
B has the more tightly clustered middle 50%.
Equal ranges do not imply equal spread in the middle.
*Bonus* Estimate the mean from this grouped frequency table:
0–4: frequency 2
5–9: frequency 4
10–14: frequency 2
Estimated mean = 7
Class midpoints: 2, 7, 12
\frac{2(2)+7(4)+12(2)}{2+4+2} =\frac{56}{8}=7
Researchers find that students who sleep longer tend to earn higher grades.
They conclude:
“Making students sleep longer will cause their grades to improve.”
What is wrong with that conclusion?
Association alone does not establish causation.
This observational finding could involve confounding variables, such as stress, work schedules, or study habits.
The study does not establish that increasing sleep causes higher grades.