Anganwadi Supervisor Paper-1 Top-ups · Chapter 4

Descriptive Statistics: Mean, Median, Mode and Variance

What to remember

Descriptive statistics summarise data. They give a central value (mean, median, mode) and a spread (range, variance, standard deviation).
Mean = sum of values / number of values. Median = middle value of ordered data. Mode = most frequent value. For moderately skewed data: Mode = 3 Median - 2 Mean.
Variance is the average of squared deviations from the mean. Standard deviation (SD) is its square root. Coefficient of variation (CV) = (SD / Mean) x 100.

Data and its presentation

Raw data: values as collected. Array: values in order.
Frequency distribution: a table of values (or class intervals) with how many times each occurs.
Class interval: a range such as 10-20. Class width = upper limit - lower limit. Mid-point = (lower + upper) / 2. Cumulative frequency: running total of frequencies.
Types of data: qualitative (categories like gender) and quantitative (numbers). Quantitative data can be discrete (counts) or continuous (measurements).
Scales: nominal, ordinal, interval and ratio.
Graphs: bar chart, pie chart, histogram (continuous data), frequency polygon, ogive (cumulative frequency curve) and line graph. The median can be read from the point where the "less than" and "more than" ogives cross.
Population is the whole group. Sample is a part chosen from it.

Measures of central tendency

1. Arithmetic mean (AM)

Ungrouped: Mean = (x1 + x2 + ... + xn) / n.
Grouped (frequency): Mean = sum(f x) / sum(f), where x is the mid-point.
Weighted mean = sum(w x) / sum(w).
Combined mean of two groups = (n1 m1 + n2 m2) / (n1 + n2).
Properties: the sum of deviations of values from the mean is zero. The mean is affected by every value, so extreme values pull it. If every value is increased by a constant, the mean rises by the same constant. If every value is multiplied by a constant, the mean is multiplied by it.
Short method: Mean = A + sum(f d) / sum(f), where A is an assumed mean and d = x - A.

2. Median

Ungrouped: arrange in order. If n is odd, take the middle value, that is the ((n + 1) / 2)th value. If n is even, take the average of the (n / 2)th and the (n / 2 + 1)th values.
Grouped: Median = L + [(N / 2 - cf) / f] x h, where L = lower limit of the median class, N = total frequency, cf = cumulative frequency before the median class, f = frequency of the median class, h = class width.
Not affected by extreme values. Good for skewed data such as income.

3. Mode

The most frequent value. A data set may have no mode, one mode (unimodal), or two (bimodal) or more.
Grouped: Mode = L + [(f1 - f0) / (2 f1 - f0 - f2)] x h, where f1 is the modal class frequency, f0 the frequency before it and f2 the frequency after it.
Only measure that can be used for qualitative data.

Empirical relation: Mode = 3 Median - 2 Mean. For a symmetrical distribution, Mean = Median = Mode.

Other means (for positive values):

Mean	Formula	Note
Arithmetic (AM)	Sum / n	Most common
Geometric (GM)	nth root of the product	Used for growth rates and ratios
Harmonic (HM)	n / sum(1 / x)	Used for rates such as speed

For positive and unequal values, AM > GM > HM. For two values a and b: GM squared = AM x HM. HM of a and b = 2ab / (a + b).

Quartiles, deciles and percentiles divide ordered data into 4, 10 and 100 equal parts. The median is Q2, D5 and P50.

Measures of dispersion (spread)

Measure	Formula
Range	Largest - Smallest
Quartile deviation	(Q3 - Q1) / 2
Mean deviation	Sum of absolute deviations from the mean (or median) / n
Variance (population)	sum((x - mean)^2) / n
Standard deviation	square root of variance
Coefficient of variation	(SD / Mean) x 100
Coefficient of range	(Max - Min) / (Max + Min)

Variance, short form: Variance = (sum of x^2 / n) - (mean)^2.
For a sample, the divisor is n - 1 for variance.
If a constant is added to every value, variance and SD do not change. If every value is multiplied by k, SD is multiplied by |k| and variance by k^2.
SD is the best-known measure of spread. CV compares the spread of two data sets with different units or means. The data with the smaller CV is more consistent.
Range is easy but uses only two values. Quartile deviation ignores extreme values.

Shape of a distribution

Skewness shows lack of symmetry. In a positively skewed distribution the tail is on the right, and Mean > Median > Mode. In a negatively skewed one the tail is on the left, and Mean < Median < Mode.
Karl Pearson's coefficient of skewness = (Mean - Mode) / SD.
Kurtosis shows peakedness. The normal curve is symmetrical and bell-shaped, with Mean = Median = Mode.

Worked examples

1. Mean. Values 4, 8, 12, 16, 20. Sum = 60. Mean = 60 / 5 = 12.
2. Median. Values 3, 7, 5, 9, 11, 1. Order: 1, 3, 5, 7, 9, 11. n = 6, so median = (5 + 7) / 2 = 6.
3. Mode. Values 2, 3, 3, 4, 5, 3, 6, 4. The value 3 occurs three times. Mode = 3.
4. Variance. Values 2, 4, 6, 8, 10. Mean = 6. Squared deviations: 16, 4, 0, 4, 16. Sum = 40. Variance = 40 / 5 = 8. SD is the square root of 8, about 2.83.
5. Combined mean. 20 students average 40, 30 students average 60. Combined = (20 x 40 + 30 x 60) / 50 = 2600 / 50 = 52.
6. Empirical relation. Mean = 40, Mode = 22. Median = (Mode + 2 x Mean) / 3 = (22 + 80) / 3 = 34.
7. Grouped mean. x = 1, 2, 3 with f = 2, 3, 5. Sum f x = 2 + 6 + 15 = 23. Sum f = 10. Mean = 2.3.
8. CV. Mean = 50, SD = 10. CV = (10 / 50) x 100 = 20 percent.
9. Grouped data. Classes 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 5, 8, 12, 10, 5. N = 40.
Mid-points: 5, 15, 25, 35, 45. Sum f x = 25 + 120 + 300 + 350 + 225 = 1020. Mean = 1020 / 40 = 25.5.
N / 2 = 20. Cumulative frequencies: 5, 13, 25, 35, 40. The median class is 20-30, with cf = 13 and f = 12. Median = 20 + (20 - 13) / 12 x 10 = 20 + 5.83 = 25.83.
Modal class is 20-30 (f1 = 12, f0 = 8, f2 = 10). Mode = 20 + (12 - 8) / (24 - 8 - 10) x 10 = 20 + 4 / 6 x 10 = 26.67.
Check: 3 x Median - 2 x Mean = 77.5 - 51 = 26.5, close to the mode found.

Use in child development work

Averages summarise weight-for-age or height-for-age of children. The median is preferred when a few children are very different. The standard deviation shows how far children differ from the average. Growth standards for children use mean and SD (Z-scores). A Z-score = (value - mean) / SD. Z-score below -2 for weight-for-age suggests underweight, and below -3 suggests severe underweight.

Exam traps

Mean is affected by extreme values. Median is not.
Mode is the only central measure usable for categories.
Adding a constant changes the mean, not the SD. Multiplying changes both.
Variance is in squared units. SD is in the original units. CV has no unit.
AM > GM > HM only for positive, unequal values.
A histogram is for continuous data. A bar chart is for separate categories.
For an even count, the median is the average of two middle values.
The sum of deviations from the mean is zero. The sum of absolute deviations is smallest about the median.

One-liners

1. Mean is the sum divided by the count.
2. Median is the middle value of ordered data.
3. Mode is the most frequent value.
4. Mode = 3 Median - 2 Mean.
5. Variance is the square of the SD.
6. CV = (SD / Mean) x 100.
7. Range = Maximum - Minimum.
8. Quartile deviation = (Q3 - Q1) / 2.
9. Q2 is the median.
10. Positive skew: Mean > Median > Mode.
11. A normal curve is bell-shaped.
12. Z-score = (value - mean) / SD.

Practice questions

The arithmetic mean of 4, 8, 12, 16 and 20 is
1. 12
2. 14
3. 10
4. 16
Answer
A. 12
Sum = 60; 60 / 5 = 12.
The median of 3, 7, 5, 9, 11 and 1 is
1. 5
2. 6
3. 7
4. 6.5
Answer
B. 6
Ordered: 1, 3, 5, 7, 9, 11; median = (5 + 7) / 2 = 6.
The mode of 2, 3, 3, 4, 5, 3, 6, 4 is
1. 2
2. 4
3. 5
4. 3
Answer
D. 3
The value 3 occurs three times, more than any other.
The range of 12, 45, 23, 9 and 31 is
1. 33
2. 36
3. 45
4. 54
Answer
B. 36
Range = 45 - 9 = 36.
The variance of 2, 4, 6, 8 and 10 (population) is
1. 6
2. 10
3. 40
4. 8
Answer
D. 8
Mean = 6; squared deviations 16, 4, 0, 4, 16 sum to 40; 40 / 5 = 8.
The mean of 10 numbers is 15. One number, 25, is replaced by 35. The new mean is
1. 16
2. 25
3. 15
4. 17
Answer
A. 16
Sum 150 becomes 160; 160 / 10 = 16.
If mean = 40 and mode = 22, the median (by the empirical relation) is
1. 31
2. 36
3. 40
4. 34
Answer
D. 34
Median = (22 + 2 x 40) / 3 = 102 / 3 = 34.
A set has mean 50 and SD 10. Its coefficient of variation is
1. 500 percent
2. 20 percent
3. 10 percent
4. 5 percent
Answer
B. 20 percent
CV = (10 / 50) x 100 = 20.
20 students average 40 marks and 30 students average 60 marks. The combined mean is
1. 52
2. 48
3. 50
4. 54
Answer
A. 52
(20 x 40 + 30 x 60) / 50 = 2600 / 50 = 52.
Marks 80 and 60 have weights 2 and 3. The weighted mean is
1. 68
2. 72
3. 70
4. 66
Answer
A. 68
(2 x 80 + 3 x 60) / 5 = 340 / 5 = 68.
The SD of a data set is 6. Every value is multiplied by 3. The new SD is
1. 6
2. 36
3. 9
4. 18
Answer
D. 18
SD is multiplied by 3, so 6 x 3 = 18.
Every value in a data set is increased by 5. The standard deviation
1. Increases by 5
2. Remains unchanged
3. Is multiplied by 5
4. Becomes zero
Answer
B. Remains unchanged
Adding a constant shifts the data but does not change the spread.
If mean = 50, mode = 44 and SD = 12, the Karl Pearson skewness is
1. -0.5
2. 0.5
3. 6
4. 0.25
Answer
B. 0.5
(50 - 44) / 12 = 0.5.
Q1 = 20 and Q3 = 40. The quartile deviation is
1. 10
2. 30
3. 20
4. 60
Answer
A. 10
(40 - 20) / 2 = 10.
Values x = 1, 2, 3 have frequencies 2, 3, 5. The mean is
1. 2.0
2. 3.0
3. 2.5
4. 2.3
Answer
D. 2.3
Sum f x = 2 + 6 + 15 = 23; 23 / 10 = 2.3.
The sum of 8 observations is 96. Their mean is
1. 10
2. 8
3. 12
4. 16
Answer
C. 12
96 / 8 = 12.
The geometric mean of 2 and 8 is
1. 5
2. 4
3. 6
4. 10
Answer
B. 4
Square root of 2 x 8 = square root of 16 = 4.
The harmonic mean of 2 and 6 is
1. 4
2. 2.5
3. 3.5
4. 3
Answer
D. 3
2 x 2 x 6 / (2 + 6) = 24 / 8 = 3.
The variance of 1, 2, 3, 4 and 5 (population) is
1. 2
2. 2.5
3. 3
4. 10
Answer
A. 2
Mean 3; squared deviations 4, 1, 0, 1, 4 sum to 10; 10 / 5 = 2.
The median of 12, 7, 15, 9 and 20 is
1. 13
2. 12
3. 9
4. 15
Answer
B. 12
Ordered: 7, 9, 12, 15, 20; the middle value is 12.
Which measure of central tendency is most affected by extreme values?
1. Quartile
2. Median
3. Mode
4. Mean
Answer
D. Mean
Every value enters the mean, so extremes pull it.
Which measure can be used for qualitative data such as favourite colour?
1. Mode
2. Mean
3. Standard deviation
4. Median
Answer
A. Mode
Mode only needs counts of categories.
The relation between the three measures for a moderately skewed distribution is
1. Mean = 3 Median - 2 Mode
2. Median = 3 Mean - 2 Mode
3. Mode = 3 Median - 2 Mean
4. Mode = 2 Median - 3 Mean
Answer
C. Mode = 3 Median - 2 Mean
This is the standard empirical relation.
For a symmetrical distribution
1. Mean < Median < Mode
2. Mean = 2 Mode
3. Mean = Median = Mode
4. Mean > Median > Mode
Answer
C. Mean = Median = Mode
In a symmetric curve all three coincide.
A comparison of the spread of two groups with different units should use the
1. Range
2. Mean
3. Mode
4. Coefficient of variation
Answer
D. Coefficient of variation
CV is unit-free.
The graph used to show cumulative frequency is the
1. Bar chart
2. Ogive
3. Histogram
4. Pie chart
Answer
B. Ogive
The ogive plots cumulative frequency against class limits.
A histogram is best used for
1. Continuous data in class intervals
2. Yes or no answers
3. Names of cities
4. Pie shares
Answer
A. Continuous data in class intervals
Histogram bars touch because classes are continuous.
A school sees that two classes have the same mean score, but class A has a smaller SD. Class A is
1. Always better
2. Not comparable
3. More consistent
4. Less consistent
Answer
C. More consistent
A smaller SD means marks are closer to the mean.
Weights of children in a village include two very heavy children. Which central value best shows the typical child?
1. Mean
2. Median
3. Range
4. Variance
Answer
B. Median
Median is not pulled by extreme values.
The sum of deviations of all values from their arithmetic mean is always
1. Equal to the SD
2. Equal to the variance
3. One
4. Zero
Answer
D. Zero
Positive and negative deviations cancel out.
The Z-score of a value is
1. (Value - mean) / SD
2. (Value - mode) x SD
3. Value / mean
4. Mean / SD
Answer
A. (Value - mean) / SD
Z-score shows how many SDs a value lies from the mean.
Consider the statements: 1. Median is not affected by extreme values. 2. Mean is affected by every value. 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 correct properties.
Consider the statements: 1. Adding a constant to every value changes the variance. 2. Multiplying every value by k multiplies the variance by k squared. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Adding a constant does not change variance.
Consider the statements: 1. In a positively skewed distribution, Mean > Median > Mode. 2. In a negatively skewed distribution, Mean < Median < 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 correct.
Consider the statements: 1. A data set can have more than one mode. 2. A data set can have no 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 possible.
Consider the statements: 1. Variance is measured in the same unit as the data. 2. Standard deviation is the square root of variance. 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 in squared units.
Consider the statements: 1. For positive unequal values, AM > GM > HM. 2. For two values, GM squared = AM x HM. 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 for two positive values.
Consider the statements: 1. Range uses all values of the data. 2. Quartile deviation ignores extreme values. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Range uses only the largest and smallest values.
Match the measure with its formula: A. Range 1. Max - Min B. Quartile deviation 2. (Q3 - Q1) / 2 C. CV 3. (SD / Mean) x 100 Which is the correct order (A-B-C)?
1. 3-2-1
2. 1-3-2
3. 2-1-3
4. 1-2-3
Answer
D. 1-2-3
Each measure is matched with its own formula.
Consider the statements: 1. Sum of absolute deviations is least when taken about the median. 2. Sum of deviations about the mean is zero. 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 properties.
Consider the statements: 1. A histogram is used for continuous data. 2. A bar chart has touching bars always. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Bars in a bar chart are separate for categories.
Consider the statements: 1. For a sample, the variance divisor is n - 1. 2. Population variance divides by n. 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 definitions.
Consider the statements: 1. Median of an even number of values is the average of the two middle values. 2. Median of an odd number of values is the middle value. 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 correct.
Consider the statements: 1. Mode = 3 Median - 2 Mean is exact for every data set. 2. It holds approximately for moderately skewed data. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
It is an empirical relation, not exact.
Consider the statements: 1. A Z-score below minus 2 for weight-for-age suggests underweight. 2. Z-score uses mean and standard deviation. 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 correct.

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1.The following data represents the weekly wages (in ₹) of the employees Find the arithmetic mean of the weekly wages (in ₹). Weekly wages 900 1000 1100 1200 1300 1400 1500 (in ₹) Number of employees 12 12 14 13 14 11 14
2.The arithmetic mean of the first 15 numbers is 82. The mean is reduced by 6 when one of the numbers is removed. Find the number that is excluded?
3.The arithmetic mean of 3, 7, 11,..., 51 is ______.
4.The arithmetic mean of 25 observations is 32.6. If the arithmetic mean of the first 24 observations is 30, what is the 25th observation?
5.The mean of the following data is 12.5. Find the missing frequency p. Data Values (x): 10, 15 Frequency (f): 5, p
6.The median of the given distribution is 28.5. Find the missing frequency x. Class 0-10 10-20 20-30 30-40 40-50 50-60 Frequency 5 x 20 15 7 5
7.The monthly incomes (in ₹ thousands) of ten employees are 35, 40, 38, 50, 46, 42, 39, 37, 41, and 44. What is the median income of the employees (in ₹ thousands)?
8.What is the arithmetic mean of the dataset {12, 25, 37, 48, 56, 61, 73, 85, 92, 104}?
9.If the arithmetic mean of five numbers is 20 and one number is changed from 10 to 20, what happens to the mean?
10.Given the data set: 12, 36, 22, 27, 31, 17. What is the median value?

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