This question has two alternatives (OR); both are answered in full below.

PART 1 - Main question: Obtain the Mean, Median and Mode of the given data.

For this distribution of 100 students' marks, Mean = 40, Median = 39.2, and Mode is approximately 36.67, calculated using the step-deviation method, interpolation formula, and modal-class formula respectively.

Step 1 - Set up the table (class, frequency f, mid-value m, deviation d=(m-A)/h with assumed mean A=35, h=10, and cumulative frequency cf):

N = Sigma(f) = 100, Sigma(f x d) = -15-14-15+0+20+30+24+20 = 50

Step 2 - Mean (step-deviation/assumed-mean method):

Mean = A + h x (Sigma(fd)/N) = 35 + 10 x (50/100) = 35 + 5 = 40

Step 3 - Median:

N/2 = 50. From the cf column, the class whose cumulative frequency first exceeds 50 is 30-40 (cf = 52), so the median class is 30-40. Here L (lower limit) = 30, cf (cumulative frequency of the class before median class) = 27, f (frequency of median class) = 25, h = 10.

Median = L + [(N/2 - cf)/f] x h = 30 + [(50-27)/25] x 10 = 30 + (23/25) x 10 = 30 + 9.2 = 39.2

Step 4 - Mode:

The highest frequency is 25, in class 30-40 - this is the modal class. L = 30, f1 (modal class frequency) = 25, f0 (frequency of class before modal class) = 15, f2 (frequency of class after modal class) = 20, h = 10.

Mode = L + [(f1-f0)/(2f1-f0-f2)] x h = 30 + [(25-15)/(2x25-15-20)] x 10 = 30 + (10/15) x 10 = 30 + 6.67 = 36.67

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Arithmetic Mean = Sum(x)/N; it has clear merits (simplicity, uses all data) and demerits (sensitivity to extreme values). The alternative problem computes AM ~ 43.43 by the step-deviation method.

What is Arithmetic Mean? Merits and demerits:

Arithmetic Mean (A.M.) is the most common mathematical average, defined as the sum of all the values in a series divided by the number of values:

A.M. (x-bar) = Sum(x) / N

For grouped/frequency data: x-bar = Sum(f*x) / Sum(f)

Merits of Arithmetic Mean:

1. Simple to understand and easy to calculate.
2. Rigidly/precisely defined by a mathematical formula - no ambiguity.
3. Based on all the observations in the series.
4. Capable of further algebraic/statistical treatment (e.g. combined mean of two groups can be computed).
5. Least affected by sampling fluctuations compared to some other measures.

Demerits of Arithmetic Mean:

1. Unduly affected by extreme (very large or very small) values/outliers.
2. Cannot be calculated for open-ended class intervals without assumptions, or for qualitative (non-numeric) data.
3. It may give a value that does not actually exist in the data (e.g. an average family size of 4.4 children).
4. Cannot be located graphically (unlike median or mode).
5. In a highly skewed distribution, the mean may not be a good representative value. …

Using the step-deviation method (assumed mean A = 7, class width h = 2), the mean of the distribution is 6.5 and the standard deviation is approximately 3.28.

Step 1 — Set up the table with class midpoints (m), the step-deviation d = (m − A)/h taking assumed mean A = 7 (midpoint of the 6–8 class) and h = 2 (class width):

Step 2 — Mean (step-deviation method):

Mean = A + h × (Σfd / N) = 7 + 2 × (−6 / 24) = 7 + 2 × (−0.25) = 7 − 0.5 = 6.5

Step 3 — Standard Deviation (step-deviation method):

σ = h × √[ (Σfd²/N) − (Σfd/N)² ]

= 2 × √[ (66/24) − (−6/24)² ]

= 2 × √[ 2.75 − 0.0625 ]

= 2 × √2.6875

= 2 × 1.6394

= 3.28 (approx.)

Check: Σfd²/N (2.75) is comfortably larger than (Σfd/N)² (0.0625), as it must be for a valid variance, and the final SD (3.28) is a reasonable fraction of the overall data range (0 to 12), confirming the computation is consistent.

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