SSC-Standard Mathematics for Forest Posts · Chapter 8

Statistics and Graphs

What to remember

Mean, median and mode are the three measures of central value. Mean = sum of values ÷ number of values. Median is the middle value of ordered data. Mode is the value that occurs most often.
For a moderately skewed distribution: Mode = 3 × Median − 2 × Mean. Spread is measured by range, quartile deviation, mean deviation, variance and standard deviation (SD).
Graphs show data in a picture: bar graph (separate bars), histogram (touching bars for continuous classes), frequency polygon, ogive (cumulative frequency curve) and pie chart (angle = value ÷ total × 360°).

Basic terms

Data are raw numbers collected in a survey, such as the girth of trees in a forest block. Frequency is the number of times a value occurs. A frequency distribution is a table of values (or classes) with their frequencies.

For grouped data, a class such as 10–20 has a lower limit 10 and an upper limit 20. The class width (size) is upper limit − lower limit = 10. The class mark (mid-value) = (lower limit + upper limit) ÷ 2 = 15. Cumulative frequency (cf) is the running total of frequencies. The range of data = highest value − lowest value.

Arithmetic mean

Ungrouped data: Mean = Σx ÷ n.

Example: heights of 5 saplings are 12, 15, 18, 20, 25 cm. Sum = 90, so mean = 90 ÷ 5 = 18 cm.

Properties of the 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.
The mean is affected by extreme values.
Combined mean of two groups = (n₁·m₁ + n₂·m₂) ÷ (n₁ + n₂).
If one new value is added, new sum = old sum + that value, and divide by the new count.

Weighted mean = Σ(w·x) ÷ Σw. Use it when values have different importance. Example: values 40, 50, 60 with weights 2, 3, 5 give (80 + 150 + 300) ÷ 10 = 53.

Grouped data

1. Direct method: Mean = Σ(f·x) ÷ Σf, where x is the class mark.
2. Assumed mean method: Mean = a + Σ(f·d) ÷ Σf, where d = x − a.
3. Step deviation method: Mean = a + h × Σ(f·u) ÷ Σf, where u = (x − a) ÷ h and h is the class width.

Example: classes 0–10, 10–20, 20–30 with f = 4, 6, 10. Class marks are 5, 15, 25. Σfx = 20 + 90 + 250 = 360. Σf = 20. Mean = 18.

Useful sums: mean of the first n natural numbers = (n + 1) ÷ 2. Sum of the first n natural numbers = n(n + 1) ÷ 2.

Median

Ungrouped: arrange in order. If n is odd, median is the ((n + 1) ÷ 2)th value. If n is even, median is the average of the (n ÷ 2)th and (n ÷ 2 + 1)th values.

Example: 3, 5, 7, 9, 11 has median 7. For 4, 8, 10, 14, 20, 22, median = (10 + 14) ÷ 2 = 12.

Grouped: Median = l + [(N ÷ 2 − cf) ÷ f] × h, where

l = lower limit of the median class (the class where cumulative frequency first reaches N ÷ 2),
N = total frequency, cf = cumulative frequency of the class before it,
f = frequency of the median class, h = class width.

Example: l = 20, N = 50, cf = 18, f = 14, h = 10. N ÷ 2 = 25. Median = 20 + (7 ÷ 14) × 10 = 25.

The median is not affected by extreme values. The ogive gives it: the "less than" and "more than" curves meet at the median.

Mode

Ungrouped: most frequent value. Data may have no mode, one mode (unimodal), two modes (bimodal) or more.

Grouped: Mode = l + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h, where l is the lower limit of the modal class (highest frequency), f₁ its frequency, f₀ the frequency of the class before it, f₂ the frequency of the class after it.

Example: l = 40, f₁ = 15, f₀ = 5, f₂ = 5, h = 10. Mode = 40 + (10 ÷ 20) × 10 = 45.

Empirical relation: Mode = 3 Median − 2 Mean. If mean = 20 and median = 18, mode = 54 − 40 = 14.

Measure	Best use	Affected by extremes?
Mean	Symmetrical numerical data	Yes
Median	Skewed data, middle position	No
Mode	Most common item or size	No

Measures of dispersion

Measure	Formula
Range	Highest − lowest
Coefficient of range	(H − L) ÷ (H + L)
Quartile deviation	(Q₃ − Q₁) ÷ 2
Interquartile range	Q₃ − Q₁
Mean deviation about the mean	Σ\	x − mean\	÷ n
Variance	Σ(x − mean)² ÷ n
Standard deviation	√Variance
Coefficient of variation	(SD ÷ Mean) × 100 per cent

Quartiles divide ordered data into four equal parts. Q₁ is the median of the lower half, Q₂ is the median, and Q₃ is the median of the upper half.

Worked example (variance): data 2, 4, 6, 8, 10. Mean = 6. Squared deviations: 16, 4, 0, 4, 16, sum 40. Variance = 40 ÷ 5 = 8. SD = √8 ≈ 2.83.

Mean deviation example: 1, 3, 5, 7. Mean 4. Absolute deviations 3, 1, 1, 3 add to 8. Mean deviation = 8 ÷ 4 = 2.

Properties of SD

SD is never negative. SD = 0 when all values are equal.
Adding a constant to every value does not change SD or variance.
Multiplying every value by k multiplies SD by |k| and variance by k².
Variance = (Σx² ÷ n) − (mean)².
A smaller CV means a more consistent (steadier) series. Example: mean 50, SD 10 gives CV = 20 per cent.

Graphs and charts

Graph	Features	Used for
Bar graph	Separate bars of equal width, gaps between them	Comparing categories (for example, species counts)
Histogram	Touching bars, class intervals on the x-axis, area proportional to frequency	Continuous grouped data
Frequency polygon	Join mid-points of the tops of histogram bars	Comparing distributions
Ogive	Plot cumulative frequency against class limits	Finding median and quartiles
Line graph	Points joined by a line	Change over time (trend)
Pie chart	Circle divided into sectors	Parts of a whole

Pie chart: central angle = (value ÷ total) × 360°. All sector angles add to 360°. For 200 saplings with 50 of teak, angle = 50 ÷ 200 × 360 = 90°. A 25 per cent share is 90°, and 50 per cent is 180°.

Reading a "less than" ogive: plot cf against the upper limit of each class. A "more than" ogive uses the lower limits and falls from left to right.

In forestry, statistics summarise tree girth, height, density per hectare, rainfall, fire incidents and wildlife counts. Charts help show trends in cover, species share and yearly change.

Exam traps

1. Histogram vs bar graph: a histogram has no gaps because classes are continuous. A bar graph has gaps.
2. Median of an even count: average the two middle values; do not pick one.
3. Order the data first before finding the median. Unordered data give wrong answers.
4. Class mark is the mid-value of a class, not its width.
5. Variance vs SD: variance is the square of SD. Do not forget the square root.
6. Mode formula: the modal class has the highest frequency, but f₀ is the class before and f₂ the class after it.
7. Adding a constant changes mean but not SD; multiplying changes both.
8. Pie chart angles: they must sum to 360°. Percentages sum to 100.

One-liners

1. Mean = Σx ÷ n.
2. Median is the middle value of ordered data.
3. Mode is the most frequent value.
4. Mode = 3 Median − 2 Mean.
5. Range = highest − lowest value.
6. Variance = Σ(x − mean)² ÷ n; SD = √variance.
7. CV = SD ÷ Mean × 100.
8. Sum of deviations from the mean is zero.
9. Pie chart sector angle = value ÷ total × 360°.
10. The two ogives meet at the median.
11. Class mark = (lower limit + upper limit) ÷ 2.
12. Frequency polygon joins the mid-points of histogram tops.

Practice questions

What is the arithmetic mean of 5, 8, 12, 15 and 20?
1. 15
2. 13
3. 10
4. 12
Answer
D. 12
Sum = 60; 60 ÷ 5 = 12.
What is the median of 3, 9, 5, 7, 11, 13, 1?
1. 8
2. 7
3. 5
4. 9
Answer
B. 7
Ordered: 1, 3, 5, 7, 9, 11, 13. The 4th value is 7.
What is the median of 4, 8, 10, 14, 20, 22?
1. 14
2. 11
3. 10
4. 12
Answer
D. 12
Even count: average of 10 and 14 = 12.
What is the mode of 2, 3, 3, 5, 5, 5, 7, 8?
1. 3
2. 5
3. 7
4. 8
Answer
B. 5
5 occurs three times, the most.
The mean of 6 numbers is 15. A seventh number 22 is added. What is the new mean?
1. 17
2. 15.5
3. 16
4. 18
Answer
C. 16
Old sum = 90; new sum = 112; 112 ÷ 7 = 16.
If the mean of a distribution is 20 and the median is 18, the mode by the empirical relation is
1. 16
2. 24
3. 22
4. 14
Answer
D. 14
Mode = 3(18) − 2(20) = 54 − 40 = 14.
If the mean is 30 and the mode is 24, the median by the empirical relation is
1. 28
2. 27
3. 29
4. 26
Answer
A. 28
Median = (Mode + 2 Mean) ÷ 3 = (24 + 60) ÷ 3 = 28.
Values 40, 50 and 60 have weights 2, 3 and 5. What is the weighted mean?
1. 52
2. 50
3. 53
4. 55
Answer
C. 53
(80 + 150 + 300) ÷ 10 = 53.
What is the range of 14, 27, 9, 31, 22?
1. 27
2. 22
3. 17
4. 31
Answer
B. 22
Range = 31 − 9 = 22.
What is the variance of 2, 4, 6, 8, 10?
1. 6
2. 10
3. 8
4. 40
Answer
C. 8
Mean 6; squared deviations 16+4+0+4+16 = 40; 40 ÷ 5 = 8.
What is the standard deviation of the data 3, 3, 3, 3?
1. 3
2. 1
3. 9
4. 0
Answer
D. 0
All values equal the mean, so every deviation is zero.
The SD of a data set is 5. Each observation is multiplied by 3. The new SD is
1. 8
2. 15
3. 5
4. 75
Answer
B. 15
SD is multiplied by the constant: 5 × 3 = 15.
If the mean is 50 and SD is 10, the coefficient of variation is
1. 20%
2. 5%
3. 10%
4. 50%
Answer
A. 20%
CV = 10 ÷ 50 × 100 = 20%.
In a pie chart of 90 units, a sector of 15 units has a central angle of
1. 60°
2. 90°
3. 45°
4. 30°
Answer
A. 60°
15 ÷ 90 × 360 = 60°.
Classes 0–10, 10–20, 20–30 have frequencies 4, 6 and 10. The mean is
1. 15
2. 17
3. 20
4. 18
Answer
D. 18
Σfx = 20 + 90 + 250 = 360; Σf = 20; mean = 18.
What is the mean of the first 15 natural numbers?
1. 7.5
2. 8
3. 7
4. 15
Answer
B. 8
Mean = (n + 1) ÷ 2 = 16 ÷ 2 = 8.
For grouped data, l = 20, N = 50, cf = 18, f = 14, h = 10. The median is
1. 24
2. 26
3. 25
4. 27
Answer
C. 25
Median = 20 + (25 − 18) ÷ 14 × 10 = 25.
For a modal class, l = 40, f₁ = 15, f₀ = 5, f₂ = 5 and h = 10. The mode is
1. 44
2. 45
3. 50
4. 47.5
Answer
B. 45
Mode = 40 + (10 ÷ 20) × 10 = 45.
The mean of 5 numbers is 40. Each number is increased by 6. The new mean is
1. 240
2. 6
3. 46
4. 40
Answer
C. 46
Adding a constant adds the same constant to the mean.
The mean of 10 numbers is 25 and the mean of 5 other numbers is 40. The mean of all 15 numbers is
1. 30
2. 32.5
3. 35
4. 28
Answer
A. 30
(250 + 200) ÷ 15 = 30.
For the ordered data 2, 4, 6, 8, 10, 12, 14, 16, the interquartile range (using the median of each half) is
1. 12
2. 6
3. 10
4. 8
Answer
D. 8
Q₁ = 5, Q₃ = 13; IQR = 13 − 5 = 8.
The mean deviation about the mean of 1, 3, 5, 7 is
1. 2
2. 1
3. 4
4. 2.5
Answer
A. 2
Mean 4; absolute deviations 3, 1, 1, 3 sum to 8; 8 ÷ 4 = 2.
Which graph has bars that touch each other because the classes are continuous?
1. Bar graph
2. Histogram
3. Pie chart
4. Line graph
Answer
B. Histogram
A histogram represents continuous class intervals without gaps.
The 'less than' and 'more than' ogives intersect at the point that gives the
1. mode
2. range
3. median
4. mean
Answer
C. median
The x-value at their intersection is the median.
Which measure is NOT affected by extreme values?
1. Range
2. Variance
3. Mean
4. Median
Answer
D. Median
The median depends only on the middle position.
A frequency polygon is drawn by joining the
1. upper limits of the classes
2. corners of the bottom of the bars
3. mid-points of the tops of histogram bars
4. lower limits of the classes
Answer
C. mid-points of the tops of histogram bars
The mid-points of the bar tops are joined by straight lines.
Which graph best shows how a forest's cover changed over several years?
1. Line graph
2. Ogive
3. Pie chart
4. Histogram of one year
Answer
A. Line graph
A line graph shows a trend over time.
Adding the same constant to every observation does not change the
1. mean
2. mode
3. median
4. standard deviation
Answer
D. standard deviation
Spread is unchanged; the whole data shift by the constant.
Which statistic is used to compare the consistency of two series?
1. Range
2. Coefficient of variation
3. Mean
4. Mode
Answer
B. Coefficient of variation
A smaller CV means a steadier series.
The sum of the deviations of all observations from their arithmetic mean is always
1. one
2. the range
3. the variance
4. zero
Answer
D. zero
Positive and negative deviations cancel exactly.
For a symmetrical distribution, the mean, median and mode are
1. equal
2. zero
3. all different
4. negative
Answer
A. equal
In a symmetrical distribution the three coincide.
The total of all sector angles in a pie chart is
1. 180°
2. 100°
3. 360°
4. 90°
Answer
C. 360°
A full circle is 360°.
Statements: 1. The mean is affected by extreme values. 2. The median is affected by extreme values. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Only the mean uses every value's size; the median uses position.
Statements: 1. Variance is the square root of the standard deviation. 2. Standard deviation is never negative. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
SD is the square root of variance; SD cannot be negative.
Statements: 1. A histogram has gaps between bars. 2. A bar graph is drawn for continuous grouped classes. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
D. Neither 1 nor 2
It is the other way round: histogram bars touch; bar graph bars have gaps.
Statements: 1. Mode is the most frequent value. 2. A data set can have more than one mode. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are true; bimodal data have two modes.
Statements: 1. Multiplying every value by 2 doubles the variance. 2. Multiplying every value by 2 doubles the SD. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Variance is multiplied by 4; SD is multiplied by 2.
Statements: 1. For even n, the median is the average of the two middle values. 2. The data must be arranged in order before finding the median. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are standard rules.
Statements: 1. In a pie chart, the angle of a sector is value ÷ total × 360°. 2. A 25 per cent share corresponds to 180°. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
25 per cent is 90°; 50 per cent is 180°.
Statements: 1. The sum of deviations of values from the median is always zero. 2. The sum of deviations of values from the mean is always zero. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Only the mean has this property.
Statements: 1. The relation Mode = 3 Median − 2 Mean holds for moderately skewed data. 2. The range uses all values of the data. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
The range uses only the highest and lowest values.
Statements: 1. Adding 10 to every observation increases the mean by 10. 2. Adding 10 to every observation increases the SD by 10. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
SD is unchanged when a constant is added.
Statements: 1. Quartiles divide ordered data into four equal parts. 2. Q₂ equals the median. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are true.
Statements: 1. An ogive is a cumulative frequency curve. 2. The median can be read from an ogive. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are true.
Statements: 1. A smaller coefficient of variation means a less consistent series. 2. CV = (SD ÷ Mean) × 100. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
A smaller CV means a more consistent series.

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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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