The frequency table shows the distribution of marks obtained by 50 learners in a Mathematics test. (a)…

Question

The frequency table shows the distribution of marks obtained by 50 learners in a Mathematics test. (a) Calculate the standard deviation.

ScanSolve AI Tutor · Accepted Answer

To calculate the standard deviation for grouped data, we first find the midpoint of each class, then calculate the mean, and finally apply the standard deviation formula.

Determine the midpoint ( $x_m$ ) for each class interval and calculate $fx_m$ and $fx_m^2$ .

We'll create an extended frequency table:

Marks ( $x$ )	Frequency ( $f$ )	Midpoint ( $x_m$ )	$fx_m$	$x_m^2$	$fx_m^2$
$10 < x \le 20$	3	15	45	225	675
$20 < x \le 30$	7	25	175	625	4375
$30 < x \le 40$	12	35	420	1225	14700
$40 < x \le 50$	15	45	675	2025	30375
$50 < x \le 60$	8	55	440	3025	24200
$60 < x \le 70$	3	65	195	4225	12675
$70 < x \le 80$	2	75	150	5625	11250

Calculate the sums for frequency, $fx_m$ , and $fx_m^2$ .
$\sum f = 3+7+12+15+8+3+2 = 50$ $\sum fx_m = 45+175+420+675+440+195+150 = 2100$ $\sum fx_m^2 = 675+4375+14700+30375+24200+12675+11250 = 98250$
Calculate the mean ( $\bar{x}$ ).
$\bar{x} = \frac{\sum fx_m}{\sum f} = \frac{2100}{50} = 42$
Calculate the standard deviation ( $\sigma$ ).
The formula for standard deviation for grouped data is:
$\sigma = \sqrt{\frac{\sum fx_m^2}{\sum f} - \left(\frac{\sum fx_m}{\sum f}\right)^2}$
Substitute the calculated values:
$\sigma = \sqrt{\frac{98250}{50} - \left(\frac{2100}{50}\right)^2}$ $\sigma = \sqrt{1965 - (42)^2}$ $\sigma = \sqrt{1965 - 1764}$ $\sigma = \sqrt{201}$ $\sigma \approx 14.1774$
Rounding to three decimal places:
$\sigma \approx 14.177$
The standard deviation is $\text{14.177}$ .

The frequency table shows the distribution of marks obtained by 50 learners in a Mathematics test. (a) Calculate the standard deviation.

Answer

Step-by-step solution

Determine the midpoint ( $x_m$ ) for each class interval and calculate $fx_m$ and $fx_m^2$ .

Calculate the sums for frequency, $fx_m$ , and $fx_m^2$ .

Calculate the mean ( $\bar{x}$ ).

Calculate the standard deviation ( $\sigma$ ).

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More Mathematics Questions

The frequency table shows the distribution of marks obtained by 50 learners in a Mathematics test. (a) Calculate the standard deviation.

Answer

Step-by-step solution

Determine the midpoint (xmx_mxm​) for each class interval and calculate fxmfx_mfxm​ and fxm2fx_m^2fxm2​.

Calculate the sums for frequency, fxmfx_mfxm​, and fxm2fx_m^2fxm2​.

Calculate the mean (xˉ\bar{x}xˉ).

Calculate the standard deviation (σ\sigmaσ).

Try one yourself

Still stuck on this one?

More Mathematics Questions

Determine the midpoint ( $x_m$ ) for each class interval and calculate $fx_m$ and $fx_m^2$ .

Calculate the sums for frequency, $fx_m$ , and $fx_m^2$ .

Calculate the mean ( $\bar{x}$ ).

Calculate the standard deviation ( $\sigma$ ).