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Calculate the mean deviation from the mean of the distribution below

Mathematics

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Answer

11.65\text{11.65}

Step-by-step solution

5 steps

To calculate the mean deviation from the mean, we'll use the formula provided: MD=∑f∣x−xˉ∣∑fMD = \frac{\sum f|x - \bar{x}|}{\sum f}.
First, we need to find the class mark (xx) for each class, then calculate the mean (xˉ\bar{x}), and finally apply the mean deviation formula.

Assumption: For the class "10-10", we will take the class mark (xx) as 10. For the other classes, the class mark is the midpoint of the interval.

Here's a table with the necessary calculations: | Class | Freq (f) | Class Mark (x) | fxfx | ∣x−xˉ∣|x - \bar{x}| | f∣x−xˉ∣f|x - \bar{x}| | | :------ | :------- | :------------- | :----- | :-------------- | :--------------- | | 10-10 | 12 | 10 | 120 | 19.32 | 231.84 | | 10-20 | 25 | 15 | 375 | 14.32 | 358.00 | | 20-30 | 32 | 25 | 800 | 4.32 | 138.24 | | 30-40 | 25 | 35 | 875 | 5.68 | 142.00 | | 40-50 | 21 | 45 | 945 | 15.68 | 329.28 | | 50-60 | 10 | 55 | 550 | 25.68 | 256.80 | | Total | ∑f=125\sum f = 125 | | ∑fx=3665\sum fx = 3665 | | ∑f∣x−xˉ∣=1456.16\sum f|x - \bar{x}| = 1456.16 |

  1. Calculate the total frequency (∑f\sum f) and the sum of fxfx (∑fx\sum fx).

    ∑f=12+25+32+25+21+10=125\sum f = 12 + 25 + 32 + 25 + 21 + 10 = 125 ∑fx=(12×10)+(25×15)+(32×25)+(25×35)+(21×45)+(10×55)\sum fx = (12 \times 10) + (25 \times 15) + (32 \times 25) + (25 \times 35) + (21 \times 45) + (10 \times 55) ∑fx=120+375+800+875+945+550=3665\sum fx = 120 + 375 + 800 + 875 + 945 + 550 = 3665
  2. Calculate the mean (xˉ\bar{x}).

    xˉ=∑fx∑f=3665125=29.32\bar{x} = \frac{\sum fx}{\sum f} = \frac{3665}{125} = 29.32
  3. Calculate the absolute deviation ∣x−xˉ∣|x - \bar{x}| for each class mark.

    For x=10x=10: ∣10−29.32∣=19.32|10 - 29.32| = 19.32
    For x=15x=15: ∣15−29.32∣=14.32|15 - 29.32| = 14.32
    For x=25x=25: ∣25−29.32∣=4.32|25 - 29.32| = 4.32
    For x=35x=35: ∣35−29.32∣=5.68|35 - 29.32| = 5.68
    For x=45x=45: ∣45−29.32∣=15.68|45 - 29.32| = 15.68
    For x=55x=55: ∣55−29.32∣=25.68|55 - 29.32| = 25.68

  4. Calculate f∣x−xˉ∣f|x - \bar{x}| for each class and sum them (∑f∣x−xˉ∣\sum f|x - \bar{x}|).

    ∑f∣x−xˉ∣=(12×19.32)+(25×14.32)+(32×4.32)+(25×5.68)+(21×15.68)+(10×25.68)\sum f|x - \bar{x}| = (12 \times 19.32) + (25 \times 14.32) + (32 \times 4.32) + (25 \times 5.68) + (21 \times 15.68) + (10 \times 25.68) ∑f∣x−xˉ∣=231.84+358.00+138.24+142.00+329.28+256.80=1456.16\sum f|x - \bar{x}| = 231.84 + 358.00 + 138.24 + 142.00 + 329.28 + 256.80 = 1456.16
  5. Calculate the Mean Deviation (MD).

    MD=∑f∣x−xˉ∣∑f=1456.16125=11.64928MD = \frac{\sum f|x - \bar{x}|}{\sum f} = \frac{1456.16}{125} = 11.64928

    Rounding to two decimal places:

    MD≈11.65MD \approx 11.65

    The mean deviation from the mean is 11.65\text{11.65}.

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