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Calculate the standard deviation for the given grouped data on the amount of money farmers have.

Mathematics

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Answer

a

177 K\text{177 K}

Step-by-step solution

3 steps

Part a· 3 steps

To calculate the standard deviation for grouped data, we first find the midpoint (xx) for each class, then calculate fxfx and fx2fx^2.

  1. Calculate midpoints (xx), fxfx, and fx2fx^2.

    Amount (K)No. of farmers (ff)Midpoint (xx)fxfxx2x^2fx2fx^2
    0<x≤1000 < x \le 100550250250012500
    100<x≤200100 < x \le 20016150240022500360000
    200<x≤300200 < x \le 300172504250625001062500
    300<x≤400300 < x \le 4001735059501225002082500
    400<x≤500400 < x \le 5001445063002025002835000
    500<x≤600500 < x \le 6001255066003025003630000
    600<x≤700600 < x \le 700765045504225002957500
    700<x≤800700 < x \le 800275015005625001125000
    Total∑f=90\sum f = 90∑fx=31800\sum fx = 31800∑fx2=14065000\sum fx^2 = 14065000

  2. Apply the formula for standard deviation (σ\sigma).

    σ=∑fx2∑f−(∑fx∑f)2\sigma = \sqrt{\frac{\sum fx^2}{\sum f} - \left(\frac{\sum fx}{\sum f}\right)^2}

    Substitute the calculated sums:

    σ=1406500090−(3180090)2\sigma = \sqrt{\frac{14065000}{90} - \left(\frac{31800}{90}\right)^2}

  3. Calculate the values.

    σ=156277.777...−(353.333...)2\sigma = \sqrt{156277.777... - (353.333...)^2} σ=156277.777...−124844.444...\sigma = \sqrt{156277.777... - 124844.444...} σ=31433.333...\sigma = \sqrt{31433.333...} σ≈177.2945\sigma \approx 177.2945

    Rounding to three significant figures:

    σ≈177 K\sigma \approx \boxed{\text{177 K}}

Part b(i)

To complete the cumulative frequency table, add the frequencies sequentially.

Amount (K)≤0\le 0≤100\le 100≤200\le 200≤300\le 300≤400\le 400≤500\le 500≤600\le 600≤700\le 700≤800\le 800
Frequency0521385569818890

Part b(ii)

To draw a smooth cumulative frequency curve

  • Plot the upper class boundaries against their corresponding cumulative frequencies. The points to plot are (0, 0), (100, 5), (200, 21), (300, 38), (400,
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