Quiz · Medium
Mean, median, standard deviation and z-scores, the empirical rule, probability rules, counting, expected value, the binomial formula and Bayes' theorem. Intro stats and AP Statistics level.
What is the median of 3, 9, 1, 7, 5 and 11?
7
5
1
6
Sort first: 1, 3, 5, 7, 9, 11. With an even count, the median is the average of the two middle values: (5 + 7) / 2 = 6. 7 is the middle of the unsorted list, a common mistake.
For a normal distribution, about what percentage of data falls within 2 standard deviations of the mean?
68%
99.7%
95%
50%
Empirical rule (68-95-99.7): about 68% within 1 SD, 95% within 2 SD and 99.7% within 3 SD.
What is the population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9?
4
5
About 2.14
2
Mean = 40 / 8 = 5. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32. Variance = 32 / 8 = 4, so . 4 is the variance, and about 2.14 is the sample standard deviation (dividing by n − 1 = 7).
Test scores have a mean of 70 and a standard deviation of 10. What is the z-score of a score of 85?
1.5
15
0.15
8.5
. The score is 1.5 standard deviations above the mean. 15 is the raw difference, before dividing by .
A data set of house prices has one mansion worth far more than the rest. Which measure of center is most resistant to this outlier?
Median
Mean
Range
Standard deviation
The median depends only on the middle position, so one extreme value barely moves it. The mean gets pulled toward the outlier, which is why incomes and home prices are usually reported as medians. Range and standard deviation measure spread, not center.
What is the mean of 4, 8, 6, 10 and 12?
6
8
10
40
Mean = sum / count = (4 + 8 + 6 + 10 + 12) / 5 = 40 / 5 = 8. 40 is the sum before dividing.
Two fair six-sided dice are rolled. What is the probability that the sum is 7?
There are 36 equally likely outcomes, and 6 give a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So . wrongly treats the 11 possible sums (2–12) as equally likely.
A fair coin is flipped 3 times. What is the probability of getting at least one head?
Use the complement: P(no heads) = , so P(at least one) = . is the probability of exactly one head.
One card is drawn from a standard 52-card deck. What is the probability it is a heart or a king?
Addition rule: . counts the king of hearts twice.
A bag has 5 red and 3 blue marbles. Two are drawn without replacement. What is the probability that both are red?
. would be correct with replacement. Without replacement, the second draw has one fewer red and one fewer marble.
Events A and B are independent, with P(A) = 0.3 and P(B) = 0.5. What is P(A and B)?
0.8
0.65
0.2
0.15
For independent events, . 0.8 is , and 0.65 is .
P(A and B) = 0.2 and P(B) = 0.5. What is P(A | B)?
0.1
0.7
0.4
2.5
. Knowing B happened shrinks the sample space to B. 0.1 multiplies instead of divides.
How many ways can a committee of 3 people be chosen from 10?
120
720
30
1,000
Order does not matter, so use combinations: . 720 is the number of ordered arrangements (permutations).
In how many ways can 4 of 6 different books be arranged in order on a shelf?
15
24
360
1,296
Order matters, so use permutations: . 15 is , which ignores order. 1,296 = allows the same book to be reused.
A game costs nothing to play. You win $10 with probability 0.2 and lose $2 otherwise. What is the expected value per game?
$3.60
$8.00
$0.40
$4.00
. On average you gain 40 cents per game. Treating the loss as a gain gives $3.60, $8.00 ignores the probabilities, and $4.00 averages the two outcomes as if they were equally likely.
A fair coin is flipped 4 times. What is the probability of getting exactly 2 heads?
Binomial: . 2 heads is the most likely outcome, but it is not 1/2, because 0, 1, 3 and 4 heads are also possible.
A disease affects 1% of people. A test detects it 99% of the time when present, and gives a false positive 5% of the time when absent. If someone tests positive, about how likely are they to have the disease?
99%
About 17%
95%
About 50%
Bayes' theorem: . Because the disease is rare, most positives come from the large healthy group. Answering 99% confuses with .
A scatterplot has a correlation coefficient r = −0.9. What does this mean?
A strong negative linear relationship
A weak negative linear relationship
No relationship, because r is negative
90% of the variation in y is caused by x
r ranges from −1 to 1. The sign shows direction and the size shows strength, so −0.9 is strong and negative. Correlation does not show causation, and the share of variation explained is , not 0.9.
Ice cream sales and drowning deaths are strongly positively correlated. What is the best explanation?
Eating ice cream causes drowning
Drowning causes people to buy ice cream
The correlation must be a calculation error
A lurking variable, hot weather, increases both
Correlation does not imply causation. Summer heat drives both swimming (and so drownings) and ice cream sales. A controlled experiment, not an observational correlation, is needed to show cause and effect.
