The table gives the frequency distribution of marks obtained by a number of students in a test. If the mean…

Question

The table gives the frequency distribution of marks obtained by a number of students in a test. If the mean mark is 5, calculate the: (a) the value of m; (b) median; (c) standard deviation of the distribution.

ScanSolve AI Tutor · Accepted Answer

Here's how to solve the problem: a) The value of m; Step 1: Write down the formula for the mean. The mean (x) is given by the sum of (mark × frequency) divided by the sum of frequencies. x = (∑ (x_i × f_i))/(∑ f_i) Step 2: Calculate the sum of (x_i × f_i). ∑ (x_i × f_i) = (3 × 5) + (4 × m) + (5 × (m+1)) + (6 × 9) + (7 × 4) + (8 × 1); ∑ (x_i × f_i) = 15 + 4m + 5m + 5 + 54 + 28 + 8; ∑ (x_i × f_i) = 110 + 9m Step 3: Calculate the sum of frequencies. ∑ f_i = 5 + m + (m+1) + 9 + 4 + 1; ∑ f_i = 20 + 2m; Step 4: Substitute the sums and the given mean (x = 5) into the formula and solve for m. 5 = (110 + 9m)/(20 + 2m) Multiply both sides by (20 + 2m): 5(20 + 2m) = 110 + 9m; 100 + 10m = 110 + 9m Subtract 9m from both sides: 10m − 9m = 110 − 100; m = 10 The value of m is 10. b) Median; Step 1: Update the frequency table with m=10. Marks (x_i) — Frequency (f_i) — Cumulative Frequency (CF)3 — 5 — 5 4 — 10 — 5+10=15; 5 — 10+1=11 — 15+11=26; 6 — 9 — 26+9=35; 7 — 4 — 35+4=39; 8 — 1 — 39+1=40 Step 2: Calculate the total number of students (N). N = ∑ f_i = 40 Step 3: Determine the position of the median. For discrete data, the median is at the ((N+1)/(2))-th position. Median position = (40+1)/(2) = 41/2 = 20.5 This means the median is the average of the 20th and 21st values. Step 4: Locate the median mark using the cumulative frequency. From the cumulative frequency column, the 20th value falls within the mark 5 category (since CF for mark 4 is 15, and CF for mark 5 is 26). The 21st value also falls within the mark 5 category. Therefore, the median is 5. The median is 5. c) Standard deviation of the distribution. Step 1: Write down the formula for standard deviation. The standard deviation (σ) for grouped data is given by: σ = √(∑ f_i x_i²)/(N) − x^2; We know N=40 and x=5. Step 2: Calculate ∑ f_i x_i². Marks (x_i) — Frequency (f_i) — x_i² — f_i x_i²3 — 5 — 9 — 5 × 9 = 45; 4 — 10 — 16 — 10 × 16 = 160; 5 — 11 — 25 — 11 × 25 = 275; 6 — 9 — 36 — 9 × 36 = 324; 7 — 4 — 49 — 4 × 49 = 196; 8 — 1 — 64 — 1 × 64 = 64 Total — 40 — 1064 So, ∑ f_i x_i² = 1064. Step 3: Substitute the values into the standard deviation formula. σ = √(1064/40 − 5²); σ = √(26.6 − 25); σ = √(1.6) σ ≈ 1.2649 Rounding to three decimal places: σ ≈ 1.265 The standard deviation is 1.265.

Marks ( $x_i$ )	Frequency ( $f_i$ )	Cumulative Frequency (CF)
3	5	5
4	10	$5+10=15$
5	$10+1=11$	$15+11=26$
6	9	$26+9=35$
7	4	$35+4=39$
8	1	$39+1=40$

Marks ( $x_i$ )	Frequency ( $f_i$ )	$x_i^2$	$f_i x_i^2$
3	5	9	$5 \times 9 = 45$
4	10	16	$10 \times 16 = 160$
5	11	25	$11 \times 25 = 275$
6	9	36	$9 \times 36 = 324$
7	4	49	$4 \times 49 = 196$
8	1	64	$1 \times 64 = 64$
Total	40		1064
So, $\sum f_i x_i^2 = 1064$ .

The table gives the frequency distribution of marks obtained by a number of students in a test. If the mean mark is 5, calculate the: (a) the value of m; (b) median; (c) standard deviation of the distribution.

Answers

Step-by-step solution

Part a· 4 steps

Write down the formula for the mean.

Calculate the sum of $(x_i \times f_i)$ .

Calculate the sum of frequencies.

Substitute the sums and the given mean ( $\bar{x} = 5$ ) into the formula and solve for $m$ .

Part b· 4 steps
$\text{5}$

Update the frequency table with $m=10$ .

Calculate the total number of students ( $N$ ).

Determine the position of the median.

Locate the median mark using the cumulative frequency.

Part c· 3 steps
$\text{1.265}$

Write down the formula for standard deviation.

Calculate $\sum f_i x_i^2$ .

Substitute the values into the standard deviation formula.

Try one yourself

Still stuck on this one?

More Mathematics Questions

The table gives the frequency distribution of marks obtained by a number of students in a test. If the mean mark is 5, calculate the: (a) the value of m; (b) median; (c) standard deviation of the distribution.

Answers

Step-by-step solution

Part a· 4 steps

Write down the formula for the mean.

Calculate the sum of (xi×fi)(x_i \times f_i)(xi​×fi​).

Calculate the sum of frequencies.

Substitute the sums and the given mean (xˉ=5\bar{x} = 5xˉ=5) into the formula and solve for mmm.

Part b· 4 steps5\text{5}5

Update the frequency table with m=10m=10m=10.

Calculate the total number of students (NNN).

Determine the position of the median.

Locate the median mark using the cumulative frequency.

Part c· 3 steps1.265\text{1.265}1.265

Write down the formula for standard deviation.

Calculate ∑fixi2\sum f_i x_i^2∑fi​xi2​.

Substitute the values into the standard deviation formula.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Calculate the sum of $(x_i \times f_i)$ .

Substitute the sums and the given mean ( $\bar{x} = 5$ ) into the formula and solve for $m$ .

Part b· 4 steps
$\text{5}$

Update the frequency table with $m=10$ .

Calculate the total number of students ( $N$ ).

Part c· 3 steps
$\text{1.265}$

Calculate $\sum f_i x_i^2$ .