The molecular weight of polyethene determined in five individual experiments is given below:

Experiment No. Molecular weight (g/mol)
1 10,000
2 11,000
3 9,000
4 10,500
5 11,500

The standard deviation in the above measurements is closest to

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  1. 850 g/mol
  2. 2000 g/mol
  3. 1600 g/mol
  4. 500 g/mol

Answer (Detailed Solution Below)

Option 1 : 850 g/mol
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Detailed Solution

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Concept:-

  • The standard deviation formula is used to find the values of a particular data that is dispersed. In simple words, the standard deviation is defined as the deviation of the values or data from an average mean.
  • A lower standard deviation concludes that the values are very close to their average.
  • Whereas higher values mean the values are far from the mean value. It should be noted that the standard deviation value can never be negative.

Standard Deviation formula (\(\sigma\)):

\(\sigma=\sqrt{\sum\frac{\left(x_i-\bar{x}\right)^2}{N-1}}\)

  • σ = Standard Deviation
  • xi = Terms Given in the Data
  • x̄ = Mean
  • n = Total number of Terms

Explanation:-

The molecular weight of polyethene determined in five individual experiments is given below:

Experiment No. Molecular weight (g/mol)
1 10,000
2 11,000
3 9,000
4 10,500
5 11,500

so, x1 = 10,000, x2 = 11,000, x3 = 9,000, x4 = 10,500, x511,500

Thus, The mean (x̄) value will be

\(\rm \bar{x}=\left(\frac{10000+11000+9000+10500+11500}{5}\right) gm/mol\)

\(=10400\; gm/mol\)

N = Total number of Terms

= 5

Now, standard deviation (s) will ne

\(\sigma=\sqrt{\sum\frac{\left(x_i-\bar{x}\right)^2}{N-1}}\)

\(\rm s=\sqrt{\frac{\left(x_1-\bar{x}\right)^2+\left(x_2-\bar{x}\right)^2+\left(x_3-\bar{x}\right)^2+\left(x_4-\bar{x}\right)^2+\left(x_5-\bar{x}\right)^2}{N-1}}\)

\(\rm s=\sqrt{\frac{\left(10000-10400\right)^2+\left(11000-10400\right)^2+\left(9000-10400\right)^2+\left(10500-10400\right)^2+\left(11500-10400\right)^2}{5-1}}\)

\(\rm s=\sqrt{\frac{\left(-400\right)^2+\left(600\right)^2+\left(-1400\right)^2+\left(100\right)^2+\left(1100\right)^2}{4}}\)

= 850 g/mol

Conclusion:-

Hence, the standard deviation in the above measurements is closest to 850 g/mol

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