ALL-IN-ONE PSTAT (PROBABILITY & STATISTICS)
EXAM QUESTIONS WITH VERIFIED ANSWERS –
LATEST 2025/2026 VERSION
300 QUESTIONS AND ANSWERS
1. :: What is the difference between a population and a sample?
Answer: A population includes all elements or individuals of interest in a study,
while a sample is a subset of the population selected for analysis.
2. :: Define the terms "parameter" and "statistic."
Answer: A parameter is a numerical characteristic of a population (e.g.,
population mean μ), while a statistic is a numerical characteristic calculated
from a sample (e.g., sample mean x̄).
3. :: What is the difference between qualitative and quantitative data?
Answer: Qualitative (categorical) data represent characteristics or qualities that
can't be numerically measured but can be classified into categories. Quantitative
data are numerical measurements that represent amounts or quantities.
4. :: Explain the difference between discrete and continuous variables.
Answer: Discrete variables can only take specific, countable values (often
integers), while continuous variables can take any value within a range and are
measured on a continuous scale.
5. :: What are the four levels of measurement in statistics?
Answer: Nominal (categories with no order), ordinal (categories with a
meaningful order), interval (ordered with equal distances but no true zero), and
ratio (ordered with equal distances and a true zero).
Measures of Central Tendency
6. :: What are the three most common measures of central tendency?
Answer: Mean (average), median (middle value), and mode (most frequent
value).
,7. :: When is the median a better measure of central tendency than the mean?
Answer: The median is better when data are skewed or contain outliers, as it's
less sensitive to extreme values.
8. :: Calculate the mean, median, and mode for the dataset: 3, 7, 8, 8, 9, 12, 15.
Answer: Mean = (3+7+8+8+9+12+15)/7 = 62/7 = 8.86; Median = 8; Mode = 8.
9. :: What is the weighted mean and when is it used?
Answer: The weighted mean is an average where some values contribute more
than others based on weights. It's used when different observations have
different levels of importance or frequency.
10. :: If a sample has a symmetric distribution, what can you say about the
relationship between the mean, median, and mode?
Answer: In a perfectly symmetric distribution, the mean, median, and mode are
all equal.
Measures of Dispersion
11. :: What are the most common measures of dispersion?
Answer: Range, variance, standard deviation, interquartile range (IQR), and
coefficient of variation.
12. :: Calculate the range, variance, and standard deviation for: 5, 7, 8, 10, 15.
Answer: Range = 15-5 = 10; Mean = 9; Variance = [(5-9)²+(7-9)²+(8-9)²+(10-
9)²+(15-9)²]/5 = (16+4+1+1+36)/5 = 58/5 = 11.6; Standard deviation = √11.6 ≈
3.41.
13. :: What information does the standard deviation provide?
Answer: The standard deviation measures how spread out values are from the
mean. A larger standard deviation indicates greater variability in the data.
14. :: Define the interquartile range (IQR) and explain when it's useful.
Answer: The IQR is the difference between the third quartile (Q₃) and the first
quartile (Q₁). It's useful for describing dispersion when data may contain
outliers, as it focuses on the middle 50% of values.
15. :: What is the coefficient of variation and when should it be used?
Answer: The coefficient of variation (CV) is the standard deviation divided by
the mean, usually expressed as a percentage. It's useful for comparing
dispersion in datasets with different units or scales.
Data Visualization
,16. :: What type of graph is best for displaying categorical data?
Answer: Bar charts, pie charts, and frequency tables are best for categorical
data.
17. :: When would you use a histogram instead of a bar chart?
Answer: Histograms are used for continuous numerical data to show the
distribution of values, while bar charts are for categorical data. Histograms have
no gaps between bars to represent the continuous nature of the data.
18. :: What is a boxplot and what information does it display?
Answer: A boxplot (box-and-whisker plot) displays the five-number summary:
minimum, Q₁ (25th percentile), median, Q₃ (75th percentile), and maximum. It
visually shows the distribution, central tendency, and potential outliers.
19. :: What type of graph would you use to display the relationship between two
continuous variables?
Answer: A scatter plot is best for showing the relationship between two
continuous variables.
20. :: What does a positively skewed distribution look like?
Answer: A positively skewed distribution has a longer tail on the right side,
with most values concentrated on the left. The mean is typically greater than the
median.
Probability
Basic Concepts
21. :: Define probability and explain the three approaches to assigning
probabilities.
Answer: Probability measures the likelihood of an event occurring, with values
between 0 and 1. The three approaches are: classical (equally likely outcomes),
relative frequency (long-run proportion), and subjective (degree of belief).
22. :: What is the difference between an experiment, outcome, and event in
probability?
Answer: An experiment is a process with a well-defined set of possible results.
An outcome is a single result of the experiment. An event is a collection of
outcomes (a subset of the sample space).
23. :: What is the sample space of an experiment?
Answer: The sample space (S) is the set of all possible outcomes of an
experiment.
, 24. :: If P(A) = 0.4 and P(B) = 0.3, what are the possible values for P(A∩B)?
Answer: The probability P(A∩B) must be between max(0, P(A)+P(B)-1) and
min(P(A), P(B)). So here, P(A∩B) must be between max(0, 0.4+0.3-1) =
max(0, -0.3) = 0 and min(0.4, 0.3) = 0.3.
25. :: What is the complement of an event and how do you calculate its
probability?
Answer: The complement of event A, denoted Aᶜ or A', is the set of all
outcomes in the sample space that are not in A. Its probability is P(Aᶜ) = 1 -
P(A).
Rules of Probability
26. :: State the addition rule of probability.
Answer: P(A∪B) = P(A) + P(B) - P(A∩B)
27. :: When can you use the simplified addition rule P(A∪B) = P(A) + P(B)?
Answer: When events A and B are mutually exclusive (A∩B = ∅), meaning
they cannot occur simultaneously.
28. :: If events A and B are independent, what is true about P(A∩B)?
Answer: For independent events, P(A∩B) = P(A) × P(B).
29. :: What is conditional probability and how is it calculated?
Answer: Conditional probability P(A|B) is the probability of event A occurring
given that event B has occurred. It's calculated as P(A|B) = P(A∩B)/P(B),
where P(B) > 0.
30. :: What does it mean when events A and B are independent?
Answer: Events A and B are independent if the occurrence of one doesn't affect
the probability of the other. Mathematically, P(A|B) = P(A) or equivalently,
P(A∩B) = P(A) × P(B).
Counting Principles
31. :: What is the multiplication principle in counting?
Answer: If operation 1 can be performed in n₁ ways, operation 2 in n₂ ways,
and so on, then the sequence of operations can be performed in n₁ × n₂ × ...
ways.
32. :: How many different 4-digit PINs are possible using the digits 0-9?
Answer: Using the multiplication principle: 10 × 10 × 10 × 10 = 10⁴ = 10,000
possible PINs.
EXAM QUESTIONS WITH VERIFIED ANSWERS –
LATEST 2025/2026 VERSION
300 QUESTIONS AND ANSWERS
1. :: What is the difference between a population and a sample?
Answer: A population includes all elements or individuals of interest in a study,
while a sample is a subset of the population selected for analysis.
2. :: Define the terms "parameter" and "statistic."
Answer: A parameter is a numerical characteristic of a population (e.g.,
population mean μ), while a statistic is a numerical characteristic calculated
from a sample (e.g., sample mean x̄).
3. :: What is the difference between qualitative and quantitative data?
Answer: Qualitative (categorical) data represent characteristics or qualities that
can't be numerically measured but can be classified into categories. Quantitative
data are numerical measurements that represent amounts or quantities.
4. :: Explain the difference between discrete and continuous variables.
Answer: Discrete variables can only take specific, countable values (often
integers), while continuous variables can take any value within a range and are
measured on a continuous scale.
5. :: What are the four levels of measurement in statistics?
Answer: Nominal (categories with no order), ordinal (categories with a
meaningful order), interval (ordered with equal distances but no true zero), and
ratio (ordered with equal distances and a true zero).
Measures of Central Tendency
6. :: What are the three most common measures of central tendency?
Answer: Mean (average), median (middle value), and mode (most frequent
value).
,7. :: When is the median a better measure of central tendency than the mean?
Answer: The median is better when data are skewed or contain outliers, as it's
less sensitive to extreme values.
8. :: Calculate the mean, median, and mode for the dataset: 3, 7, 8, 8, 9, 12, 15.
Answer: Mean = (3+7+8+8+9+12+15)/7 = 62/7 = 8.86; Median = 8; Mode = 8.
9. :: What is the weighted mean and when is it used?
Answer: The weighted mean is an average where some values contribute more
than others based on weights. It's used when different observations have
different levels of importance or frequency.
10. :: If a sample has a symmetric distribution, what can you say about the
relationship between the mean, median, and mode?
Answer: In a perfectly symmetric distribution, the mean, median, and mode are
all equal.
Measures of Dispersion
11. :: What are the most common measures of dispersion?
Answer: Range, variance, standard deviation, interquartile range (IQR), and
coefficient of variation.
12. :: Calculate the range, variance, and standard deviation for: 5, 7, 8, 10, 15.
Answer: Range = 15-5 = 10; Mean = 9; Variance = [(5-9)²+(7-9)²+(8-9)²+(10-
9)²+(15-9)²]/5 = (16+4+1+1+36)/5 = 58/5 = 11.6; Standard deviation = √11.6 ≈
3.41.
13. :: What information does the standard deviation provide?
Answer: The standard deviation measures how spread out values are from the
mean. A larger standard deviation indicates greater variability in the data.
14. :: Define the interquartile range (IQR) and explain when it's useful.
Answer: The IQR is the difference between the third quartile (Q₃) and the first
quartile (Q₁). It's useful for describing dispersion when data may contain
outliers, as it focuses on the middle 50% of values.
15. :: What is the coefficient of variation and when should it be used?
Answer: The coefficient of variation (CV) is the standard deviation divided by
the mean, usually expressed as a percentage. It's useful for comparing
dispersion in datasets with different units or scales.
Data Visualization
,16. :: What type of graph is best for displaying categorical data?
Answer: Bar charts, pie charts, and frequency tables are best for categorical
data.
17. :: When would you use a histogram instead of a bar chart?
Answer: Histograms are used for continuous numerical data to show the
distribution of values, while bar charts are for categorical data. Histograms have
no gaps between bars to represent the continuous nature of the data.
18. :: What is a boxplot and what information does it display?
Answer: A boxplot (box-and-whisker plot) displays the five-number summary:
minimum, Q₁ (25th percentile), median, Q₃ (75th percentile), and maximum. It
visually shows the distribution, central tendency, and potential outliers.
19. :: What type of graph would you use to display the relationship between two
continuous variables?
Answer: A scatter plot is best for showing the relationship between two
continuous variables.
20. :: What does a positively skewed distribution look like?
Answer: A positively skewed distribution has a longer tail on the right side,
with most values concentrated on the left. The mean is typically greater than the
median.
Probability
Basic Concepts
21. :: Define probability and explain the three approaches to assigning
probabilities.
Answer: Probability measures the likelihood of an event occurring, with values
between 0 and 1. The three approaches are: classical (equally likely outcomes),
relative frequency (long-run proportion), and subjective (degree of belief).
22. :: What is the difference between an experiment, outcome, and event in
probability?
Answer: An experiment is a process with a well-defined set of possible results.
An outcome is a single result of the experiment. An event is a collection of
outcomes (a subset of the sample space).
23. :: What is the sample space of an experiment?
Answer: The sample space (S) is the set of all possible outcomes of an
experiment.
, 24. :: If P(A) = 0.4 and P(B) = 0.3, what are the possible values for P(A∩B)?
Answer: The probability P(A∩B) must be between max(0, P(A)+P(B)-1) and
min(P(A), P(B)). So here, P(A∩B) must be between max(0, 0.4+0.3-1) =
max(0, -0.3) = 0 and min(0.4, 0.3) = 0.3.
25. :: What is the complement of an event and how do you calculate its
probability?
Answer: The complement of event A, denoted Aᶜ or A', is the set of all
outcomes in the sample space that are not in A. Its probability is P(Aᶜ) = 1 -
P(A).
Rules of Probability
26. :: State the addition rule of probability.
Answer: P(A∪B) = P(A) + P(B) - P(A∩B)
27. :: When can you use the simplified addition rule P(A∪B) = P(A) + P(B)?
Answer: When events A and B are mutually exclusive (A∩B = ∅), meaning
they cannot occur simultaneously.
28. :: If events A and B are independent, what is true about P(A∩B)?
Answer: For independent events, P(A∩B) = P(A) × P(B).
29. :: What is conditional probability and how is it calculated?
Answer: Conditional probability P(A|B) is the probability of event A occurring
given that event B has occurred. It's calculated as P(A|B) = P(A∩B)/P(B),
where P(B) > 0.
30. :: What does it mean when events A and B are independent?
Answer: Events A and B are independent if the occurrence of one doesn't affect
the probability of the other. Mathematically, P(A|B) = P(A) or equivalently,
P(A∩B) = P(A) × P(B).
Counting Principles
31. :: What is the multiplication principle in counting?
Answer: If operation 1 can be performed in n₁ ways, operation 2 in n₂ ways,
and so on, then the sequence of operations can be performed in n₁ × n₂ × ...
ways.
32. :: How many different 4-digit PINs are possible using the digits 0-9?
Answer: Using the multiplication principle: 10 × 10 × 10 × 10 = 10⁴ = 10,000
possible PINs.
