Mathematics Classes VI-X for School Assistants and TET Paper 2A · Chapter 11

Statistics: Frequency Tables, Central Tendency and Ogives

What to remember

Mean, median and mode are the three measures of central tendency. For grouped data: Mean = Σfx / Σf; Median = l + [(n/2 − cf) / f] × h; Mode = l + [(f1 − f0) / (2f1 − f0 − f2)] × h.
Empirical relation: Mode = 3 Median − 2 Mean (for moderately skewed data).
Ogive is a cumulative frequency graph. The "less than" and "more than" ogives cut each other at a point whose x-value is the median.

Data and frequency tables

Data means collected facts or numbers. Raw data is data in the form it was first collected. Range = highest value − lowest value.

A frequency is the number of times a value occurs. A frequency distribution table lists values with their frequencies. Tally marks are used while counting; every fifth tally crosses the four before it.

For large data we make class intervals (groups). Some terms:

Class limits: the lowest and highest values of a class (lower limit, upper limit).
Class size (width) h: upper limit − lower limit.
Class mark (mid-value): (lower limit + upper limit) / 2.
Inclusive classes: 10–19, 20–29. Both end values belong to the class. There is a gap between classes.
Exclusive classes: 10–20, 20–30. The upper limit is not in the class (it goes to the next class). There is no gap.

To convert inclusive to exclusive classes, subtract half of the gap from each lower limit and add half of the gap to each upper limit. For 10–19, 20–29 the gap is 1, so the classes become 9.5–19.5, 19.5–29.5.

Cumulative frequency (cf) of a class is the sum of the frequencies of that class and all classes before it. The last cf equals n = Σf.

Mean

Ungrouped data: Mean = (sum of all values) / (number of values). Example: 5, 7, 9, 11, 13 give 45 / 5 = 9.

Properties of the arithmetic mean:

The sum of deviations of all values from the mean is zero.
If every value is increased by k, the mean increases by k. If every value is multiplied by k, the mean is multiplied by k.
It is affected by extreme values.

Grouped data: use the class mark x of each class.

1. Direct method: Mean = Σfx / Σf.
2. Assumed mean method: take an assumed mean a, find d = x − a. Mean = a + Σfd / Σf.
3. Step deviation method: u = (x − a) / h. Mean = a + h × Σfu / Σf.

Worked example. Classes 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 5, 8, 15, 16, 6 (Σf = 50).

Class	f	x	fx	cf
0–10	5	5	25	5
10–20	8	15	120	13
20–30	15	25	375	28
30–40	16	35	560	44
40–50	6	45	270	50

Σfx = 1350, so Mean = 1350 / 50 = 27. Check by step deviation with a = 25, h = 10: u = −2, −1, 0, 1, 2; Σfu = −10 − 8 + 0 + 16 + 12 = 10; Mean = 25 + 10 × 10 / 50 = 27. Both methods agree.

Median

The median is the middle value when data is arranged in order. It divides the data into two equal parts.

Ungrouped, n odd: the ((n + 1)/2)th value.
Ungrouped, n even: the average of the (n/2)th and (n/2 + 1)th values. Example: 3, 5, 8, 10 have median (5 + 8)/2 = 6.5.
Grouped data: find n/2, locate the median class (the first class whose cf is at least n/2). Then

Median = l + [(n/2 − cf) / f] × h

where l = lower limit of the median class, cf = cumulative frequency of the class before it, f = frequency of the median class, h = class size.

In the example above n/2 = 25. The cf column shows 28 as the first value ≥ 25, so the median class is 20–30. Here l = 20, cf = 13, f = 15, h = 10. Median = 20 + (25 − 13)/15 × 10 = 20 + 8 = 28.

The median is not affected by extreme values, so it suits skewed data such as income.

Mode

The mode is the value that occurs most often. Data can have no mode, one mode (unimodal) or more than one mode (bimodal, multimodal). Example: 2, 3, 3, 5, 3, 7 has mode 3.

For grouped data, the modal class is the class with the highest frequency. Then

Mode = l + [(f1 − f0) / (2f1 − f0 − f2)] × h

where l = lower limit of the modal class, f1 = its frequency, f0 = frequency of the class before it, f2 = frequency of the class after it, h = class size.

In the example the modal class is 30–40 (f1 = 16, f0 = 15, f2 = 6). Mode = 30 + (16 − 15) / (32 − 15 − 6) × 10 = 30 + 10/11 ≈ 30.91.

Empirical relation: Mode = 3 Median − 2 Mean. With Median = 28 and Mean = 27 we get Mode = 84 − 54 = 30, close to 30.91. This shows the formula is an approximation.

Measure	Best used when	Affected by extreme values?
Mean	all values matter; data is fairly even	Yes
Median	data is skewed or has outliers	No
Mode	most common item is needed (shoe size, shirt size)	No

Ogives (cumulative frequency curves)

An ogive is a graph of cumulative frequency.

Less than ogive

Make a table of "less than" cumulative frequencies against the upper limit of each class.
Points for the example: (10, 5), (20, 13), (30, 28), (40, 44), (50, 50). Start the curve at (0, 0), the lower limit of the first class with cf 0.
The curve rises from left to right.

More than ogive

Use "more than" cumulative frequencies against the lower limit of each class.
Points: (0, 50), (10, 45), (20, 37), (30, 22), (40, 6). End at (50, 0).
The curve falls from left to right.

Finding the median from ogives. Draw both curves on the same axes. They meet at one point. Draw a perpendicular from that point to the x-axis; the foot is the median. Another way: mark n/2 on the y-axis, draw a horizontal line to the less than ogive, then drop a perpendicular to the x-axis.

Scale choice matters. Use equal scales on each axis and mark the zero point clearly. Join the points by a smooth free-hand curve, not by ruler.

Classroom angle

Start with data from the class itself (heights, number of siblings, marks). Students collect, tabulate and then find the average.
Use a bar graph or histogram before moving to ogives. A histogram has no gaps between bars for exclusive classes; a bar graph has equal gaps.
Let students see that the mean can be a number that no one actually has (for example 4.6 children), but the mode is always an actual observed value.

Exam traps

Median class is found by n/2, not (n + 1)/2, for grouped data.
"cf" in the median formula is the cf of the class before the median class, not of the median class itself.
Less than ogive uses the upper limit; more than ogive uses the lower limit.
In the mode formula, f0 is the frequency before and f2 the frequency after the modal class. Swapping them gives a wrong answer.
Class mark is the mid-value, not the class size.
Inclusive classes must first be converted to exclusive form before applying median or ogive formulas.
Mean of the means of groups is correct only when the groups are of equal size; otherwise use weights.
The sum of deviations from the mean is zero, but the sum of deviations from the median is not zero in general.

One-liners

1. Range = maximum value − minimum value.
2. Class mark = (lower limit + upper limit) / 2.
3. Class size for 10–20 is 10.
4. The last cumulative frequency equals the total frequency n.
5. Mean = Σfx / Σf.
6. Step deviation uses u = (x − a) / h.
7. Median class contains the (n/2)th item.
8. Mode = 3 Median − 2 Mean (empirical).
9. The median is the x-value where the two ogives intersect.
10. The sum of deviations from the mean is zero.
11. Adding 5 to every value adds 5 to the mean.
12. For a perfectly symmetric distribution, mean, median and mode are equal.

Practice questions

What is the class mark of the class 20-30?
1. 30
2. 20
3. 25
4. 10
Answer
C. 25
Class mark = (20 + 30)/2 = 25.
What is the range of the data 4, 9, 2, 15, 7?
1. 15
2. 9
3. 11
4. 13
Answer
D. 13
Range = 15 - 2 = 13.
For grouped data, the mean by the direct method is given by
1. Σf / Σfx
2. Σfx / Σf
3. Σx / Σf
4. Σfx / n²
Answer
B. Σfx / Σf
Mean = sum of (frequency × class mark) divided by total frequency.
The median class of grouped data is the class that contains the
1. class with the largest class mark
2. (n/2)th item by cumulative frequency
3. first class
4. item with highest frequency
Answer
B. (n/2)th item by cumulative frequency
The median class is the first class whose cumulative frequency is at least n/2.
The x-coordinate of the point where the less than ogive and the more than ogive intersect gives the
1. mode
2. range
3. median
4. mean
Answer
C. median
The two ogives cross at the median.
A less than ogive is drawn by plotting cumulative frequencies against the
1. upper limits of the classes
2. lower limits of the classes
3. class sizes
4. class marks
Answer
A. upper limits of the classes
Less than cumulative frequency refers to values below the upper limit.
A more than ogive is drawn by plotting cumulative frequencies against the
1. class marks
2. frequencies
3. upper limits of the classes
4. lower limits of the classes
Answer
D. lower limits of the classes
More than cumulative frequency refers to values at or above the lower limit.
What is the mode of 2, 3, 3, 5, 3, 7?
1. 2
2. 5
3. 4.5
4. 3
Answer
D. 3
3 occurs three times, more than any other value.
The empirical relation among the three measures of central tendency is
1. Mode = 3 Median - 2 Mean
2. Mode = Mean + Median
3. Mode = 2 Median - 3 Mean
4. Mode = 3 Mean - 2 Median
Answer
A. Mode = 3 Median - 2 Mean
For moderately skewed data, Mode = 3 Median - 2 Mean.
The sum of the deviations of all values from their arithmetic mean is always
1. one
2. zero
3. equal to n
4. equal to the mean
Answer
B. zero
Positive and negative deviations cancel exactly.
Which measure of central tendency is NOT affected by extreme values?
1. Arithmetic mean
2. Sum of values
3. Median
4. Range
Answer
C. Median
The median depends only on the middle position, so outliers do not change it.
The last value in the cumulative frequency column is equal to the
1. mean
2. largest class mark
3. total frequency n
4. class size
Answer
C. total frequency n
Cumulative frequency adds up all frequencies.
The classes 10-19 and 20-29 are converted into exclusive classes. The common boundary is
1. 20.5
2. 19
3. 20
4. 19.5
Answer
D. 19.5
The gap is 1, so half-gap 0.5 is added to 19 and subtracted from 20.
The modal class of a grouped distribution is the class with the
1. highest class mark
2. highest frequency
3. highest cumulative frequency
4. lowest frequency
Answer
B. highest frequency
Modal class has the maximum frequency.
The median of 3, 5, 8, 10 is
1. 6.5
2. 6
3. 8
4. 5
Answer
A. 6.5
For even n, median = average of middle values (5 + 8)/2 = 6.5.
The mean of the first five natural numbers is
1. 2.5
2. 5
3. 15
4. 3
Answer
D. 3
(1+2+3+4+5)/5 = 15/5 = 3.
The graph of cumulative frequencies is called
1. a histogram
2. an ogive
3. a pie chart
4. a frequency polygon
Answer
B. an ogive
A cumulative frequency curve is an ogive.
The mean of 5, 7, 9, 11, 13 is
1. 45
2. 10
3. 9
4. 8
Answer
C. 9
Sum = 45, n = 5, so mean = 9.
Classes 0-10, 10-20, 20-30, 30-40, 40-50 have frequencies 5, 8, 15, 16, 6. The mean of the distribution is
1. 27
2. 25
3. 30
4. 28
Answer
A. 27
Σfx = 25 + 120 + 375 + 560 + 270 = 1350; 1350/50 = 27.
Classes 0-10, 10-20, 20-30, 30-40, 40-50 have frequencies 5, 8, 15, 16, 6. The median of the distribution is
1. 28
2. 30
3. 25
4. 27
Answer
A. 28
n/2 = 25, median class 20-30: 20 + (25-13)/15 × 10 = 28.
Classes 0-10, 10-20, 20-30, 30-40, 40-50 have frequencies 5, 8, 15, 16, 6. The modal class and the mode are respectively
1. 40-50 and about 41
2. 30-40 and exactly 35
3. 20-30 and about 25.9
4. 30-40 and about 30.9
Answer
D. 30-40 and about 30.9
Modal class has f = 16; mode = 30 + (16-15)/(32-15-6) × 10 ≈ 30.91.
The mean of five numbers is 20. If 3 is added to each number, the new mean is
1. 20
2. 60
3. 23
4. 17
Answer
C. 23
Adding a constant k to each value adds k to the mean.
The mean of 6 numbers is 8. The sum of five of them is 38. The sixth number is
1. 12
2. 10
3. 8
4. 9
Answer
B. 10
Total = 48; 48 - 38 = 10.
For a distribution, mean = 4 and median = 5. Using the empirical formula, the mode is
1. 6
2. 3
3. 4.5
4. 7
Answer
D. 7
Mode = 3 × 5 - 2 × 4 = 7.
The mean of x, x + 2 and x + 4 is 10. The value of x is
1. 6
2. 8
3. 7
4. 10
Answer
B. 8
Mean = x + 2 = 10, so x = 8.
In a step deviation method a = 50, h = 10, Σfu = -6 and Σf = 30. The mean is
1. 48
2. 44
3. 52
4. 47
Answer
A. 48
Mean = 50 + 10 × (-6)/30 = 48.
In a distribution with n = 40, the median class is 20-30, the cumulative frequency before it is 14 and its frequency is 12. The median is
1. 27.5
2. 24
3. 25
4. 26
Answer
C. 25
20 + (20 - 14)/12 × 10 = 20 + 5 = 25.
The modal class is 30-35 with f1 = 20, the previous class has frequency 12 and the next class has frequency 8. The mode is
1. 33
2. 31
3. 32
4. 32.5
Answer
C. 32
Mode = 30 + (20-12)/(40-12-8) × 5 = 30 + 8/20 × 5 = 32.
The median of 7, 3, 9, 1, 5, 11, 13 is
1. 9
2. 6
3. 5
4. 7
Answer
D. 7
Sorted: 1, 3, 5, 7, 9, 11, 13. Middle (4th) value is 7.
The average mark of 40 students is 50 and that of another 10 students is 60. The average of all 50 students is
1. 52
2. 50
3. 54
4. 55
Answer
A. 52
(2000 + 600)/50 = 52.
The mean of the data 2, 4, 6 with frequencies 3, 4, 3 respectively is
1. 3
2. 4.5
3. 4
4. 6
Answer
C. 4
Σfx = 6 + 16 + 18 = 40 and Σf = 10, so mean = 4.
Consider the statements. 1. The arithmetic mean is affected by extreme values. 2. The median is not affected by extreme values. Which is/are correct?
1. 1 only
2. 2 only
3. Neither 1 nor 2
4. Both 1 and 2
Answer
D. Both 1 and 2
Both statements are true.
Consider the statements. 1. A less than ogive rises from left to right. 2. A more than ogive rises from left to right. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
A more than ogive falls from left to right.
Consider the statements. 1. A data set may have no mode. 2. A data set may have more than one mode. Which is/are correct?
1. 1 only
2. Both 1 and 2
3. 2 only
4. Neither 1 nor 2
Answer
B. Both 1 and 2
Data with all different values has no mode; two or more equal peaks give bimodal/multimodal data.
Consider the statements. 1. The class mark of 10-20 is 15. 2. The class size of 10-20 is 5. Which is/are correct?
1. 2 only
2. Both 1 and 2
3. 1 only
4. Neither 1 nor 2
Answer
C. 1 only
Class size is 20 - 10 = 10.
Consider the statements. 1. The sum of deviations from the mean is zero. 2. The sum of deviations from the median is always zero. Which is/are correct?
1. 2 only
2. Both 1 and 2
3. Neither 1 nor 2
4. 1 only
Answer
D. 1 only
Only the mean has this property in general.
Consider the statements. 1. A histogram of continuous classes has no gaps between bars. 2. A bar graph has no gaps between bars. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Bar graphs have equal gaps between the bars.
Consider the statements. 1. For a symmetric distribution, mean, median and mode are equal. 2. Mode = 3 Mean - 2 Median. Which is/are correct?
1. 2 only
2. 1 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 1 only
The correct relation is Mode = 3 Median - 2 Mean.
Consider the statements. 1. In a median formula, cf is the cumulative frequency of the median class. 2. In a median formula, cf is the cumulative frequency of the class preceding the median class. Which is/are correct?
1. 1 only
2. Both 1 and 2
3. 2 only
4. Neither 1 nor 2
Answer
C. 2 only
The cf in the formula is that of the class before the median class.
Consider the statements. 1. The cumulative frequency of the first class equals its own frequency. 2. The cumulative frequency of the last class equals n. Which is/are correct?
1. 1 only
2. 2 only
3. Neither 1 nor 2
4. Both 1 and 2
Answer
D. Both 1 and 2
Cumulative frequency is a running total starting from the first class.
Consider the statements. 1. If every value is multiplied by 4, the mean is multiplied by 4. 2. If every value is increased by 4, the mean is multiplied by 4. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Adding 4 to each value adds 4 to the mean.
Consider the statements. 1. The class mark is the difference between the class limits. 2. The mode is the middle value of the arranged data. Which is/are correct?
1. 1 only
2. Neither 1 nor 2
3. 2 only
4. Both 1 and 2
Answer
B. Neither 1 nor 2
Class mark is the mid-value; the middle value is the median.
Match the measure with its formula. P. Mean Q. Median R. Mode 1. l + [(f1 - f0)/(2f1 - f0 - f2)] × h 2. Σfx/Σf 3. l + [(n/2 - cf)/f] × h
1. P-1, Q-3, R-2
2. P-2, Q-3, R-1
3. P-3, Q-2, R-1
4. P-2, Q-1, R-3
Answer
B. P-2, Q-3, R-1
Mean is Σfx/Σf, median uses n/2 and cf, mode uses f1, f0, f2.
Match the term with its meaning. P. Class mark Q. Range R. Class size 1. Upper limit - lower limit of a class 2. Highest value - lowest value 3. Mid-value of a class
1. P-1, Q-2, R-3
2. P-2, Q-3, R-1
3. P-3, Q-2, R-1
4. P-3, Q-1, R-2
Answer
C. P-3, Q-2, R-1
Class mark is the mid-value, range is max - min, class size is the width of a class.
The two ogives of a distribution are drawn on the same axes. The point where they meet is (28, 25) for n = 50. What does the value 28 represent?
1. Median
2. Range
3. Mode
4. Mean
Answer
A. Median
The x-coordinate of the intersection is the median; 25 = n/2.

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1.The median of the observations 31, 40, 74, 47, 14, 44, 46, 29, 36, 97 and 87 is:
2.The median of the observations 87, 56, 27, 31, 13, 39, 18, 80, 98, 92 and 25 is:
3.The median of the observations 60, 43, 24, 61, 72, 26, 39, 62, 49, 95 and 38 is:
4.The median of the observations 34, 58, 98, 42, 83, 62, 12, 11, 87, 75 and 15 is:
5.The median of the observations 87, 56, 79, 81, 11, 53, 94, 45, 32, 99 and 98 is:
6.The mode of the observations 9, 6, 4, 2, 5, 2, 9, 2, 1, 9, 4, 5, 9, 5, 4 and 8 is:
7.The mode of the numbers 7, 8, 7, 9, 10 is:
8.The mode of the observations 8, 7, 4, 7, 9, 4, 5, 4, 7, 2, 4, 5, 8, 5, 4 and 5 is:
9.The mode of the observations 4, 1, 5, 2, 8, 6, 7, 4, 3, 5, 1, 6, 4, 9 and 4 is:
10.The mode of the observations 35, 32, 28, 27, 25, 32, 31, 27, 32, 30, 32, 27, 31, 34, 32 and 22 is:

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