Statistics and probability quiz

Question 1

What is the difference between the mean and the median of a dataset, and which is more robust to outliers?

Accepted Answer

The median is more robust to outliers than the mean

Answer

The mean is always smaller than the median; the mean is more robust

Answer

They are always equal; both equally robust

Answer

The median is the average; the mean is the middle value

Question 2

In a machine learning dataset, why is variance an important measure when describing a feature?

Accepted Answer

Variance measures the spread of values around the mean

Answer

Variance measures the average value of a feature

Answer

Variance is always equal to the mean

Answer

A zero-variance feature is the most informative

Question 3

What is the relationship between standard deviation and variance?

Accepted Answer

Standard deviation is the square root of the variance

Answer

Standard deviation is the square of the variance

Answer

They are always equal

Answer

Variance is the square root of standard deviation

Question 4

A bag has 3 red and 7 blue balls. What is the probability of drawing a red ball at random?

Accepted Answer

0.3

Answer

0.7

Answer

0.5

Answer

0.21

Question 5

State the addition rule for the probability of A OR B for two events that may overlap.

Accepted Answer

P(A) + P(B) − P(A ∩ B)

Answer

P(A) + P(B)

Answer

P(A) × P(B)

Answer

P(A) − P(B) + P(A ∩ B)

Question 6

What does it mean for two events to be statistically independent, and how is it expressed mathematically?

Accepted Answer

P(A ∩ B) = P(A) × P(B)

Answer

P(A ∩ B) = P(A) + P(B)

Answer

P(A | B) = P(B)

Answer

P(A ∪ B) = P(A) × P(B)

Question 7

State Bayes' theorem and explain what each part represents.

Accepted Answer

P(A|B) = [P(B|A) P(A)] / P(B)

Answer

P(A|B) = P(A)P(B) / P(B|A)

Answer

P(A|B) = P(B|A) + P(A) − P(B)

Answer

P(A|B) = P(B|A) / [P(A) P(B)]

Question 8

A disease affects 1% of a population. A test is 99% accurate (both sensitivity and specificity). If someone tests positive, roughly what is the probability they actually have the disease?

Accepted Answer

About 50%

Answer

About 99%

Answer

About 1%

Answer

About 90%

Question 9

What is conditional probability, and how is P(A | B) calculated?

Accepted Answer

P(A | B) = P(A ∩ B) / P(B)

Answer

P(A | B) = P(A) × P(B)

Answer

P(A | B) = P(A ∩ B) / P(A)

Answer

P(A | B) = P(A) + P(B)

Question 10

What is the expected value (expectation) of a discrete random variable, and what does it represent?

Accepted Answer

The probability-weighted average of outcomes

Answer

The most likely single outcome

Answer

The largest possible value

Answer

The middle value of the distribution

Question 11

What are the parameters of a binomial distribution, and what does it model?

Accepted Answer

Number of trials n and success probability p

Answer

Mean μ and standard deviation σ

Answer

Rate λ only

Answer

Lower and upper bounds a and b

Question 12

Describe the key properties of the normal (Gaussian) distribution.

Accepted Answer

Symmetric bell shape defined by μ and σ

Answer

Skewed right, defined by rate λ

Answer

Uniform and flat between two bounds

Answer

Discrete with only integer values

Question 13

According to the empirical (68-95-99.7) rule, what percentage of normally distributed data falls within two standard deviations of the mean?

Accepted Answer

95%

Answer

68%

Answer

99.7%

Answer

50%

Question 14

What is a z-score, and why is it useful in data preprocessing?

Accepted Answer

z = (x − μ) / σ, used for standardization

Answer

z = x × σ + μ, used for scaling up

Answer

z = (x − median) / range

Answer

z = μ / σ, the coefficient of variation

Question 15

What does the Pearson correlation coefficient measure, and what is its range?

Accepted Answer

Strength and direction of linear relationship, range −1 to 1

Answer

Strength of any relationship, range 0 to 1

Answer

Causation strength, range 0 to 100

Answer

Variance ratio, range −∞ to ∞

Question 16

Explain the saying 'correlation does not imply causation' with a brief example.

Accepted Answer

A correlation may be driven by a hidden confounder

Answer

Correlation always proves cause and effect

Answer

Causation is weaker than correlation

Answer

Correlated variables can never be related

Question 17

What is covariance, and how does it differ from correlation?

Accepted Answer

Covariance is unstandardized; correlation is standardized covariance

Answer

Covariance is always between −1 and 1

Answer

They are identical measures

Answer

Correlation has units; covariance is unitless

Question 18

State the Central Limit Theorem (CLT) and why it matters in data science.

Accepted Answer

Sample means become approximately normal as sample size grows

Answer

All data becomes normal as you collect more points

Answer

The population mean always equals zero

Answer

Variance disappears with large samples

Question 19

What is the difference between a population and a sample in statistics?

Accepted Answer

A sample is a subset of the population

Answer

A population is a subset of a sample

Answer

They are the same thing

Answer

A sample is always larger than the population

Question 20

Why do we divide by (n−1) instead of n when computing the sample variance?

Accepted Answer

To correct bias from using the sample mean (Bessel's correction)

Answer

To make the variance larger for no reason

Answer

Because n−1 is easier to compute

Answer

To force the variance to equal the standard deviation

Question 21

What is the mode of a dataset, and when is it especially useful?

Accepted Answer

The most frequently occurring value

Answer

The average value of the data

Answer

The middle value when sorted

Answer

The range of the data

Question 22

What is skewness, and what does a positively (right) skewed distribution look like?

Accepted Answer

Long tail to the right, mean greater than median

Answer

Symmetric with mean equal to median

Answer

Long tail to the left, mean less than median

Answer

Perfectly uniform with no tails

Question 23

What is the interquartile range (IQR), and how is it used to detect outliers?

Accepted Answer

IQR = Q3 − Q1, used to flag outliers beyond 1.5×IQR

Answer

IQR = max − min, used to find the mean

Answer

IQR = Q2, the median

Answer

IQR = standard deviation × 2

Question 24

What does a p-value represent in hypothesis testing?

Accepted Answer

The probability of data this extreme if the null hypothesis is true

Answer

The probability the null hypothesis is true

Answer

The probability of making a correct decision

Answer

The size of the effect being measured

Question 25

In hypothesis testing, what is the difference between a Type I and a Type II error?

Accepted Answer

Type I rejects a true null; Type II fails to reject a false null

Answer

Type I rejects a false null; Type II rejects a true null

Answer

Both mean accepting the null hypothesis

Answer

Type I is always worse than Type II

Question 26

What is a confidence interval, and how should a 95% confidence interval be interpreted?

Accepted Answer

95% of such intervals (over repeated sampling) contain the true parameter

Answer

95% of the data lies in the interval

Answer

The parameter has a 95% chance of being the sample mean

Answer

The interval contains 95% of future samples

Question 27

What is the probability mass function (PMF) of a Poisson distribution used to model?

Accepted Answer

Number of events in a fixed interval at constant rate λ

Answer

Number of successes in a fixed number of trials

Answer

A continuous symmetric bell curve

Answer

Time between two consecutive failures

Question 28

A fair coin is tossed 3 times. What is the probability of getting exactly 2 heads?

Accepted Answer

3/8

Answer

1/8

Answer

1/2

Answer

2/3

Question 29

What is the law of large numbers, and how does it relate to estimating probabilities?

Accepted Answer

Observed frequency converges to true probability as trials increase

Answer

Large samples always have zero variance

Answer

Probabilities increase with more trials

Answer

The mean grows without bound for large samples

Question 30

What is a probability distribution, and what is the difference between discrete and continuous distributions?

Accepted Answer

Discrete assigns probabilities to countable outcomes; continuous spreads it over ranges

Answer

Discrete uses density functions; continuous uses mass functions

Answer

Both assign nonzero probability to exact values

Answer

There is no real difference between them

Test yourself