Strengthen your data skills with this free PMMA statistics reviewer and practice test. Statistics questions ask you to summarize data, compare values, and judge how likely an event is to happen.
This page reviews measures of central tendency and spread, frequency tables, probability, and counting methods, with worked examples for each. These are independently prepared learning materials, not official Philippine Merchant Marine Academy examination questions. Use these exercises as supplementary Mathematics practice, rather than as a guaranteed preview of the actual examination.
PMMA Statistics Practice Test
Work out each problem on scratch paper before choosing an answer. For data sets, sort the values from least to greatest before you begin.
Read every table or graph title, label, and scale carefully before computing anything.
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What Is Statistics?
Statistics is the study of collecting, organizing, analyzing, and interpreting data. A few basic terms appear throughout this reviewer:
- A population is the entire group being studied, such as all cadets in an academy.
- A sample is a smaller group taken from the population, such as 50 cadets chosen at random.
- Qualitative data describe categories, such as favorite subject. Quantitative data are numbers, such as test scores.
- Quantitative data can be discrete, meaning counted, such as the number of crew members, or continuous, meaning measured, such as height or temperature.
Most examples on this page use one small data set. These are the scores of eight cadets on a 10-item quiz:
2, 4, 4, 4, 5, 5, 7, 9
Statistics Topics to Practice
1. Mean
The mean, or average, is the sum of all values divided by the number of values.
Example: Find the mean of the quiz scores.
- Add the scores: 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40
- Divide by the number of scores: 40 ÷ 8 = 5
The relationship mean × number of values = total also solves missing-value problems.
Example: The mean of five numbers is 12. Four of them are 10, 11, 14, and 15. Find the fifth number.
- Total of all five: 12 × 5 = 60
- Total of the four known numbers: 10 + 11 + 14 + 15 = 50
- The fifth number is 60 − 50 = 10
2. Median
The median is the middle value after the data are sorted. With an odd number of values, it is the single middle value. With an even number, it is the mean of the two middle values.
Example: Find the median of 3, 8, 6, 10, and 5.
- Sort the values: 3, 5, 6, 8, 10
- The middle value is 6
Example: Find the median of the quiz scores.
- The eight sorted scores have two middle values, the 4th and 5th: 4 and 5
- Median: (4 + 5) ÷ 2 = 4.5
The median is less affected by extreme values than the mean. If the top score of 9 were replaced by 90, the mean would jump to about 15.1, but the median would stay at 4.5.
3. Mode
The mode is the value that appears most often. A data set may have one mode, more than one mode, or no mode at all.
Example: In the quiz scores, 4 appears three times, more than any other value. The mode is 4.
4. Weighted Mean
A weighted mean gives some values more importance than others. Multiply each value by its weight, add the products, and divide by the total weight.
Example: A student earns 90 in Math (5 units), 85 in English (3 units), and 80 in Science (2 units). Find the weighted average.
- Multiply each grade by its units: 90 × 5 = 450, 85 × 3 = 255, 80 × 2 = 160
- Add the products: 450 + 255 + 160 = 865
- Divide by the total units: 865 ÷ 10 = 86.5
5. Range and Standard Deviation
Measures of spread show how far the data are scattered. The range is the highest value minus the lowest value. For the quiz scores, the range is 9 − 2 = 7.
The standard deviation shows how far values typically lie from the mean. A small standard deviation means the data are clustered closely together.
Example: Find the standard deviation of the quiz scores, treating the eight cadets as the whole group.
- Start with the mean: 5
- Subtract the mean from each score: −3, −1, −1, −1, 0, 0, 2, 4
- Square each difference: 9, 1, 1, 1, 0, 0, 4, 16
- Add the squares: 32
- Divide by the number of values to get the variance: 32 ÷ 8 = 4
- Take the square root: standard deviation = 2
When the data are a sample rather than a whole population, divide by n − 1 instead. Here that gives a variance of 32 ÷ 7 ≈ 4.57 and a standard deviation of about 2.14.
Two shortcuts are worth remembering. Adding the same number to every value shifts the mean, median, and mode by that number but leaves the range and standard deviation unchanged. Multiplying every value by a positive number multiplies the mean, range, and standard deviation by that number.
6. Frequency Tables
A frequency table shows how many times each value occurs. To find the mean, multiply each value by its frequency, add the products, and divide by the total frequency.
Example: The scores of 20 cadets on a 10-item quiz are summarized below.
| Score (x) | Frequency (f) | f × x |
|---|---|---|
| 6 | 2 | 12 |
| 7 | 5 | 35 |
| 8 | 8 | 64 |
| 9 | 4 | 36 |
| 10 | 1 | 10 |
| Total | 20 | 157 |
- Mean: 157 ÷ 20 = 7.85
- Mode: the score with the highest frequency is 8
- Median: the first 7 cadets scored 6 or 7 and the next 8 scored 8, so the 10th and 11th scores are both 8. The median is 8
7. Probability
Probability measures how likely an event is. It ranges from 0 (impossible) to 1 (certain).
- Basic probability: P(event) = number of favorable outcomes ÷ total number of outcomes
- Complement: P(not A) = 1 − P(A)
- Independent events: P(A and B) = P(A) × P(B)
- Mutually exclusive events: P(A or B) = P(A) + P(B)
Example: A bag holds 5 red, 3 blue, and 2 green marbles.
- P(blue) = 3 ÷ 10 = 3/10
- P(not green) = 1 − 2/10 = 4/5
- P(two red marbles in a row, without replacement) = 5/10 × 4/9 = 20/90 = 2/9
In the last case, the second draw has one fewer red marble and one fewer marble in total.
Example: A coin is tossed and a die is rolled. Find the probability of getting heads and a 6.
- The events are independent: 1/2 × 1/6 = 1/12
8. Permutations and Combinations
The fundamental counting principle says that if one choice can be made in m ways and a second in n ways, both can be made in m × n ways. With 3 shirts and 4 pairs of pants, there are 3 × 4 = 12 outfits.
Factorial notation multiplies a number by every positive whole number below it: 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.
- Permutations count arrangements where order matters: ₙPᵣ = n! ÷ (n − r)!
- Combinations count selections where order does not matter: ₙCᵣ = n! ÷ [r!(n − r)!]
Example: In how many ways can a captain, a first mate, and a navigator be chosen from 6 cadets?
- Each role is different, so order matters: ₆P₃ = 6 × 5 × 4 = 120
Example: In how many ways can a 3-member committee be chosen from 6 cadets?
- All members have the same role, so order does not matter: ₆C₃ = ₆P₃ ÷ 3! = 120 ÷ 3! = 120 ÷ 6 = 20
How to Solve Statistics Questions
Sort the data first. Ordering the values makes the median, mode, and range easy to find and helps you spot errors.
Count the values carefully. Many mistakes come from miscounting n, especially when values repeat.
Use totals for missing values. When a problem gives a mean and asks for an unknown value, multiply the mean by the number of values to get the total.
Define the outcomes in probability problems. Identify the total number of outcomes, the favorable outcomes, and whether items are replaced after each draw.
Ask whether order matters. If swapping the positions creates a different result, use permutations. If not, use combinations.
Read graphs and tables closely. Check the title, labels, units, and scale before reading off any value.
Common Mistakes to Avoid
One frequent mistake is finding the median without sorting the data first. The middle value of an unsorted list is usually not the median.
Another mistake is forgetting to average the two middle values when a data set has an even number of values.
Learners also treat a weighted mean as a simple mean. When values carry different weights, such as subjects with different units, each value must be multiplied by its weight first.
Finally, check that every probability falls between 0 and 1. An answer greater than 1 signals an error, often a total that was not reduced after drawing without replacement.
Make Statistics Practice Part of Your Review
Copy the formulas from this reviewer onto a single sheet and review them until you can recall each one without looking. Then practice with small data sets of your own, such as your weekly expenses or quiz scores.
Record the types of problems you miss most often, such as standard deviation or combinations, and practice those until the method feels natural.
For additional English, Mathematics, Science, and reasoning exercises, visit our PMMA entrance exam reviewer and coverage guide.