📊 Mathematics — Statistics and Probability

1. In statistics, what is the main purpose of a frequency distribution table?

1. To arrange raw data into classes and show the number of times each class occurs
2. To arrange data in order of magnitude only
3. To show only the average of a set of data
4. To display data using angles in a circle

A frequency distribution table organises raw data into classes (or values) alongside the frequency (number of times) each class occurs. (JAMB UTME Mathematics Syllabus, Section V(1): Representation of data — frequency distribution tables)

2. Which of the following is true of a histogram used to represent continuous grouped data with equal class widths?

1. The bars are separated by equal gaps
2. The bars are drawn as adjacent rectangles with no gaps between them
3. Only the vertical axis carries numerical values
4. Each bar represents a single raw score

Because grouped continuous data has no gaps between classes, histogram bars are drawn touching one another, unlike a bar chart. (JAMB UTME Mathematics Syllabus, Section V(1) — interpreting information on a histogram)

3. In a bar chart used to represent discrete data, how are the bars usually drawn?

1. Touching one another with no gaps
2. As circular sectors
3. Separated from one another by equal gaps
4. Joined together by a smooth curve

A bar chart represents distinct discrete categories, so its bars are drawn with equal gaps between them. (JAMB UTME Mathematics Syllabus, Section V(1) — interpreting information on a bar chart)

4. The sector angles in a pie chart always add up to how many degrees?

1. 180°
2. 90°
3. 100°
4. 360°

A pie chart represents a full circle, so the sum of all its sector angles is 360°, with each sector proportional to the frequency it represents. (Standard SSCE/UTME mathematics (New General Mathematics); required for pie-chart interpretation in JAMB Syllabus Section V(1))

5. A survey of 60 students' favourite drinks gave the results: Tea = 12, Coffee = 18, Juice = 6, Water = 24. What is the sector angle representing Juice on a pie chart of this data?

1. 24°
2. 36°
3. 72°
4. 108°

Angle = (frequency ÷ total frequency) × 360° = (6 ÷ 60) × 360° = 36°. (Standard SSCE/UTME mathematics — pie-chart sector angle formula (JAMB Syllabus Section V(1)))

6. In a grouped frequency distribution, the class mark (or class midpoint) of a class interval is best described as?

1. The difference between the upper and lower class limits
2. The average of the lower and upper limits (or boundaries) of the class
3. The highest frequency in the distribution
4. The total number of observations in the class

The class mark is the midpoint of a class interval, obtained by averaging its lower and upper limits (or boundaries). (Standard SSCE/UTME mathematics (New General Mathematics); JAMB Syllabus Section V(1))

7. What is the class mark of the class interval 41 – 50?

1. 45
2. 45.5
3. 46
4. 50.5

Class mark = (lower limit + upper limit) ÷ 2 = (41 + 50) ÷ 2 = 45.5. (Standard SSCE/UTME mathematics — definition of class mark (JAMB Syllabus Section V(1)))

8. In a frequency table with class intervals 20 – 29, 30 – 39, 40 – 49, and so on, what is the upper class boundary of the interval 20 – 29?

1. 29
2. 29.5
3. 30
4. 20.5

Since the consecutive class limits 29 and 30 differ by 1, the class boundary lies halfway between them, giving an upper class boundary of 29.5 for the class 20–29. (Standard SSCE/UTME mathematics — class boundaries in grouped data (JAMB Syllabus Section V(1)))

9. What is meant by the 'class width' (or class size) of a grouped frequency distribution?

1. The number of items in the class
2. The difference between the upper and lower class boundaries of the class
3. The frequency of the modal class
4. The midpoint of the class

Class width is the size of a class interval, found by subtracting its lower class boundary from its upper class boundary. (Standard SSCE/UTME mathematics; JAMB Syllabus Section V(1))

10. The table shows marks scored by 40 students: 1–10 (5 students), 11–20 (8), 21–30 (15), 31–40 (7), 41–50 (5). What is the modal class?

1. 1 – 10
2. 11 – 20
3. 21 – 30
4. 31 – 40

The modal class is the class interval with the highest frequency; here 21–30 has the highest frequency, 15. (Standard SSCE/UTME mathematics — modal class from a frequency table (JAMB Syllabus Section V(1)-(2)))

11. A frequency polygon is obtained from a histogram by?

1. Joining the midpoints of the tops of the bars with straight lines
2. Joining the upper class boundaries with a smooth curve
3. Shading alternate bars of the histogram
4. Plotting cumulative frequency against class boundary

A frequency polygon is formed by joining the midpoints of the tops of adjacent histogram bars using straight lines. (Standard SSCE/UTME mathematics — frequency polygon construction (JAMB Syllabus Section V(1)))

12. The cumulative frequency of a class in a frequency distribution is best defined as?

1. The frequency of that class only
2. The sum of the frequencies of that class and all classes before it
3. The average frequency of all the classes
4. The frequency of the last class in the table

Cumulative frequency is a running total obtained by adding the frequency of each class to the sum of the frequencies of all preceding classes. (JAMB UTME Mathematics Syllabus, Section V(2) — use of cumulative frequency/ogive to find median, quartiles and percentiles)

13. A frequency distribution has class frequencies 4, 9, 13, 8, 6 in order. What is the cumulative frequency of the third class?

1. 13
2. 22
3. 26
4. 34

Cumulative frequency of the third class = sum of the first three frequencies = 4 + 9 + 13 = 26. (Standard SSCE/UTME mathematics — computing cumulative frequency (JAMB Syllabus Section V(2)))

14. A frequency distribution of the ages of 100 workers has cumulative frequencies 8, (8 + x), 45, 78 and 100 for five successive classes in that order. If the frequency of the third class is 20, find the value of x.

1. 12
2. 17
3. 20
4. 25

The frequency of the third class equals its cumulative frequency minus that of the second class: 45 − (8 + x) = 20, so 37 − x = 20 and x = 17. (Standard SSCE/UTME mathematics — relationship between cumulative frequency and class frequency (JAMB Syllabus Section V(2)))

15. What is an ogive?

1. A bar chart showing discrete frequencies
2. A graph of cumulative frequency plotted against the upper class boundary of each class
3. A pie chart showing percentages
4. A table listing class limits only

An ogive (cumulative frequency curve) is obtained by plotting cumulative frequency against the upper class boundary of each class and joining the points with a smooth curve. (JAMB UTME Mathematics Syllabus, Section V(2) — use of the ogive to find median, quartiles and percentiles)

16. Which description best fits the typical shape of an ogive?

1. A closed circular shape
2. A set of separate straight-line segments with no curve
3. A smooth, rising S-shaped curve from a cumulative frequency of 0 up to the total frequency
4. A straight line through the origin with constant gradient

An ogive is a smooth, rising S-shaped curve because cumulative frequency starts at 0 and increases steadily up to the total frequency, N. (Standard SSCE/UTME mathematics — shape of the cumulative frequency curve (JAMB Syllabus Section V(2)))

17. On an ogive, the first point plotted (before any class has contributed its frequency) corresponds to?

1. A cumulative frequency of 0 at the lower class boundary of the first class
2. The total frequency N at the lower class boundary of the first class
3. A cumulative frequency of 0 at the upper class boundary of the last class
4. The modal frequency at the class mark of the first class

Before the first class contributes any observations, the cumulative frequency is 0, and this point is plotted at the lower class boundary of the first class. (Standard SSCE/UTME mathematics — plotting an ogive (JAMB Syllabus Section V(2)))

18. To estimate the median of grouped data with total frequency N from an ogive, a student should?

1. Read off the value on the horizontal axis corresponding to a cumulative frequency of N/2
2. Read off the value on the horizontal axis corresponding to a cumulative frequency of N
3. Read off the cumulative frequency corresponding to the class mark of the modal class
4. Divide N by the number of classes and read that value directly

The median corresponds to the middle position of the data, so one locates N/2 on the cumulative frequency axis and reads the corresponding value on the horizontal axis. (JAMB UTME Mathematics Syllabus, Section V(2) — 'use ogive to find the median, quartiles and percentiles')

19. Using an ogive for data with total frequency N, the lower quartile (Q1) is estimated by locating which cumulative frequency on the vertical axis?

1. N/2
2. N/4
3. 3N/4
4. N

The lower quartile Q1 marks the point below which a quarter of the observations lie, so it is read off at a cumulative frequency of N/4 on the ogive. (JAMB UTME Mathematics Syllabus, Section V(2) — use of ogive to find quartiles)

20. Using an ogive, the upper quartile (Q3) of a distribution of total frequency N is read off at a cumulative frequency of?

1. N/4
2. N/2
3. 3N/4
4. N

The upper quartile Q3 is the value below which three-quarters of the data lie, so it corresponds to a cumulative frequency of 3N/4 on the ogive. (JAMB UTME Mathematics Syllabus, Section V(2) — use of ogive to find quartiles)

21. From an ogive drawn for 80 students' scores, the value corresponding to a cumulative frequency of 20 is 22 marks, and the value corresponding to a cumulative frequency of 60 is 38 marks. What is the interquartile range of the distribution?

1. 8
2. 16
3. 22
4. 38

Since N/4 = 20 gives Q1 = 22 and 3N/4 = 60 gives Q3 = 38, the interquartile range Q3 − Q1 = 38 − 22 = 16. (Standard SSCE/UTME mathematics — interquartile range from an ogive (JAMB Syllabus Section V(2)-(3)))

22. From an ogive drawn for the heights of 120 plants, the value at a cumulative frequency of 30 is 15 cm and the value at a cumulative frequency of 90 is 27 cm. Calculate the semi-interquartile range of the distribution.

1. 3 cm
2. 6 cm
3. 12 cm
4. 24 cm

Q1 = 15 cm (at N/4 = 30) and Q3 = 27 cm (at 3N/4 = 90); semi-interquartile range = (Q3 − Q1) ÷ 2 = (27 − 15) ÷ 2 = 6 cm. (Standard SSCE/UTME mathematics — semi-interquartile range from an ogive (JAMB Syllabus Section V(2)-(3)))

23. An ogive is drawn for the ages of 200 applicants. To estimate the 70th percentile of the distribution from the ogive, which cumulative frequency should be located on the vertical axis?

1. 70
2. 100
3. 140
4. 200

The kth percentile is located at a cumulative frequency of (k/100) × N; for k = 70 and N = 200, this is (70/100) × 200 = 140. (JAMB UTME Mathematics Syllabus, Section V(2) — use of ogive to find percentiles)

24. In the JAMB UTME Mathematics syllabus, interpreting bar charts, pie charts and histograms is a required skill under which sub-topic of Statistics?

1. Measures of dispersion
2. Representation of data
3. Permutation and combination
4. Probability

Section V(1) of the syllabus, 'Representation of data', requires candidates to interpret frequency tables, histograms, bar charts and pie charts. (JAMB UTME Mathematics Syllabus, Section V(1): Representation of data)

25. What is the sum of all the sector angles in a pie chart?

1. 180°
2. 270°
3. 360°
4. 400°

A pie chart represents a whole circle, so all sector angles must add up to 360°. (Standard SSCE/UTME mathematics; pie-chart interpretation, JAMB Syllabus Section V(1))

26. A family's monthly budget is shown on a pie chart, with Food taking up 40% of the budget. What angle on the pie chart represents Food?

1. 144°
2. 100°
3. 120°
4. 160°

Angle = (percentage/100) × 360° = 0.40 × 360° = 144°. (Angle of sector = (frequency/total) × 360°, standard pie-chart rule, JAMB Syllabus Section V(1))

27. In a survey of 720 students, 150 chose Physics as their favourite subject. If this information is drawn on a pie chart, what angle represents the students who chose Physics?

1. 60°
2. 75°
3. 70°
4. 80°

Angle = (150/720) × 360° = 75°, applying the sector-angle formula for pie charts. (Angle of sector = (frequency/total frequency) × 360°, JAMB Syllabus Section V(1))

28. A bar chart shows the number of items sold over five days: Monday 20, Tuesday 35, Wednesday 15, Thursday 25, Friday 30. What percentage of the total weekly sales occurred on Tuesday?

1. 28%
2. 24%
3. 26%
4. 30%

Total sales = 125 units; Tuesday's share = (35/125) × 100% = 28%. (Bar chart interpretation, JAMB Syllabus Section V(1): Representation of data)

29. Which feature distinguishes a histogram from an ordinary bar chart when representing grouped data?

1. A histogram can only display percentages
2. A histogram requires the data to be arranged in a circle
3. A histogram must be drawn in a single colour
4. A histogram has bars that touch each other because it represents continuous class intervals

Because histogram class intervals are continuous, adjacent bars share boundaries and are drawn touching, unlike bar charts for discrete categories which have gaps. (Standard SSCE/UTME mathematics; histogram interpretation, JAMB Syllabus Section V(1))

30. A histogram is drawn from a frequency table with equal class widths: 10–14 (frequency 6), 15–19 (frequency 10), 20–24 (frequency 14), 25–29 (frequency 8). Which class is the modal class?

1. 20–24
2. 10–14
3. 15–19
4. 25–29

The modal class is the class interval with the highest frequency, which is 20–24 with a frequency of 14. (JAMB Syllabus Section V(2): mode as highest-frequency class, applied via histogram, Section V(1))

31. When the class intervals of a histogram have unequal widths, what should the height of each bar represent so that the area of the bar correctly reflects the frequency of that class?

1. The raw frequency of the class
2. The cumulative frequency up to that class
3. The midpoint of the class interval
4. The frequency density, that is, frequency divided by class width

With unequal class widths, height must equal frequency density (frequency ÷ class width) so that bar area, not height, is proportional to frequency. (Standard SSCE/UTME mathematics extension of histogram interpretation, JAMB Syllabus Section V(1))

32. A pie chart is generally the most suitable chart for showing which of the following?

1. The proportion of each category relative to the whole
2. The trend of a variable over successive time periods
3. The exact class boundaries of continuous data
4. The relationship between two independent continuous variables

A pie chart splits a circle into sectors whose sizes show how each category contributes proportionally to the whole. (Standard SSCE/UTME mathematics; pie-chart interpretation, JAMB Syllabus Section V(1))

33. A pie chart shows the colours of cars sold at a dealership: Red 90°, Blue 120°, Black 100°, and White makes up the remaining sector. What angle represents White?

1. 40°
2. 45°
3. 55°
4. 50°

Since all sector angles sum to 360°, White = 360° − (90° + 120° + 100°) = 50°. (Sector angles in a pie chart sum to 360°, JAMB Syllabus Section V(1))

34. A pie chart shows how 180 students are distributed among four subjects: Mathematics (100°), English (80°), Biology (90°), and Chemistry (the remaining sector). How many students offer Chemistry?

1. 35
2. 45
3. 40
4. 50

Chemistry's angle = 360° − (100°+80°+90°) = 90°; number of students = (90/360) × 180 = 45. (Sector angle formula for pie charts, JAMB Syllabus Section V(1))

35. Which formula correctly gives the number of permutations of n distinct objects taken r at a time?

1. nPr = n!/r!
2. nPr = (n − r)!/n!
3. nPr = n!/(n − r)!
4. nPr = n!/(r!(n − r)!)

Permutations count ordered arrangements, given by nPr = n!/(n − r)!. (JAMB UTME Mathematics Syllabus, Section V(4): Permutation and combination)