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A class of 100 students' marks are given in the table. Find the mean and median mark.

Mathematics

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Answer

a
27.45
27.5

Step-by-step solution

4 steps

Here's the solution for question 2 from Section B.

2 (i) Class of 100 students' marks

Part a· 4 steps

Find the mean and median mark.

  1. Calculate the sum of (mark ×\times frequency) and total frequency for the mean.

    Let xx be the mark and ff be the frequency.

    ∑f=1+2+5+12+30+28+15+5+2=100\sum f = 1+2+5+12+30+28+15+5+2 = 100 ∑fx=(5×1)+(10×2)+(15×5)+(20×12)+(25×30)+(30×28)+(35×15)+(40×5)+(45×2)\sum fx = (5 \times 1) + (10 \times 2) + (15 \times 5) + (20 \times 12) + (25 \times 30) + (30 \times 28) + (35 \times 15) + (40 \times 5) + (45 \times 2) ∑fx=5+20+75+240+750+840+525+200+90=2745\sum fx = 5 + 20 + 75 + 240 + 750 + 840 + 525 + 200 + 90 = 2745

  2. Calculate the mean.

    Mean=∑fx∑f=2745100=27.45\text{Mean} = \frac{\sum fx}{\sum f} = \frac{2745}{100} = 27.45

  3. Construct a cumulative frequency table to find the median.

    Mark (xx)Frequency (ff)Cumulative Frequency (CF)
    511
    1023
    1558
    201220
    253050
    302878
    351593
    40598
    452100

  4. Determine the median.

    Since the total number of students is N=100N=100 (an even number), the median is the average of the N2\frac{N}{2}-th and (N2+1)(\frac{N}{2}+1)-th values.
    The 1002=50\frac{100}{2} = 50-th value.
    The (1002+1)=51(\frac{100}{2}+1) = 51-st value.
    From the cumulative frequency table:
    The 50th value is 25 (as CF reaches 50 at mark 25).
    The 51st value is 30 (as CF goes from 50 to 78 at mark 30).

    Median=50th value+51st value2=25+302=552=27.5\text{Median} = \frac{50\text{th value} + 51\text{st value}}{2} = \frac{25 + 30}{2} = \frac{55}{2} = 27.5

    The mean mark is 27.45\boxed{27.45} and the median mark is 27.5\boxed{27.5}.

Part b

Construct the cumulative frequency table of distribution.

The cumulative frequency table is shown in Step 3 of part (a).

Mark (xx)Frequency (ff)Cumulative Frequency (CF)
511
1023
1558
201220
253050
302878
351593
40598
452100

Part c

On a graph paper, by taking 2cm to represent 5 units on the x-axis and 2cm for 10 units on the y-axis construct the cumulative frequency curve of this distribution.

To construct the cumulative frequency curve (ogive):

  1. Draw the x-axis (horizontal) for 'Mark' and the y-axis (vertical) for 'Cumulative Frequency'.
  2. Scale the x-axis: Mark 0, 5, 10, 15, ..., 45. Each 2cm represents 5 units.
  3. Scale the y-axis: Cumulative Frequency 0, 10, 20, ..., 100. Each 2cm represents 10 units.
  4. Plot the points using the (Mark, Cumulative Frequency) pairs from the table: (0, 0) - (This is a common starting point for an ogive) (5, 1) (10, 3) (15, 8) (20, 20) (25, 50) (30, 78) (35, 93) (40, 98) (45, 100)
  5. Draw a smooth curve connecting these plotted points. The curve should be S-shaped.
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