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Statistics - Standard deviation of a discrete frequency distribution

Grade 11CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A discrete frequency distribution consists of distinct values x1,x2,…,xnx_1, x_2, \dots, x_n occurring with frequencies f1,f2,…,fnf_1, f_2, \dots, f_n respectively.

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The Mean (xˉ\bar{x}) of such a distribution is calculated as xˉ=∑i=1nfixiN\bar{x} = \frac{\sum_{i=1}^{n} f_i x_i}{N}, where N=∑i=1nfiN = \sum_{i=1}^{n} f_i is the total frequency.

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Variance (σ2\sigma^2) is the arithmetic mean of the squares of deviations of all items from their arithmetic mean, weighted by their frequencies.

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Standard Deviation (σ\sigma) is the positive square root of the variance.

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The Shortcut Method or Assumed Mean Method involves taking an assumed mean AA and calculating deviations di=xi−Ad_i = x_i - A. This is particularly useful when data values are large.

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Standard deviation is independent of the change of origin but dependent on the change of scale.

📐Formulae

xˉ=∑fixiN\bar{x} = \frac{\sum f_i x_i}{N}

σ=∑fi(xi−xˉ)2N\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{N}}

σ=∑fixi2N−(∑fixiN)2\sigma = \sqrt{\frac{\sum f_i x_i^2}{N} - \left(\frac{\sum f_i x_i}{N}\right)^2}

σ=∑fidi2N−(∑fidiN)2 where di=xi−A\sigma = \sqrt{\frac{\sum f_i d_i^2}{N} - \left(\frac{\sum f_i d_i}{N}\right)^2} \text{ where } d_i = x_i - A

σ=1NN∑fidi2−(∑fidi)2\sigma = \frac{1}{N} \sqrt{N \sum f_i d_i^2 - (\sum f_i d_i)^2}

💡Examples

Problem 1:

Find the variance and standard deviation for the following discrete frequency distribution: xix_i: 2, 4, 6, 8, 10 fif_i: 1, 2, 3, 2, 1

Solution:

  1. Calculate N=∑fiN = \sum f_i: 1232+19\begin{array}{r} 1 \\ 2 \\ 3 \\ 2 \\ + 1 \\ \hline 9 \end{array} So, N=9N = 9.

  2. Calculate the Mean (xˉ\bar{x}): ∑fixi=(1×2)+(2×4)+(3×6)+(2×8)+(1×10)\sum f_i x_i = (1 \times 2) + (2 \times 4) + (3 \times 6) + (2 \times 8) + (1 \times 10) 281816+1054\begin{array}{r} 2 \\ 8 \\ 18 \\ 16 \\ + 10 \\ \hline 54 \end{array} xˉ=549=6\bar{x} = \frac{54}{9} = 6.

  3. Calculate deviations (xi−xˉ)(x_i - \bar{x}) and fi(xi−xˉ)2f_i(x_i - \bar{x})^2:

  • For x=2:1(2−6)2=1(−4)2=16x=2: 1(2-6)^2 = 1(-4)^2 = 16
  • For x=4:2(4−6)2=2(−2)2=8x=4: 2(4-6)^2 = 2(-2)^2 = 8
  • For x=6:3(6−6)2=3(0)2=0x=6: 3(6-6)^2 = 3(0)^2 = 0
  • For x=8:2(8−6)2=2(2)2=8x=8: 2(8-6)^2 = 2(2)^2 = 8
  • For x=10:1(10−6)2=1(4)2=16x=10: 1(10-6)^2 = 1(4)^2 = 16

  1. Sum of fi(xi−xˉ)2f_i(x_i - \bar{x})^2: 16808+1648\begin{array}{r} 16 \\ 8 \\ 0 \\ 8 \\ + 16 \\ \hline 48 \end{array}

  2. Variance (σ2\sigma^2): σ2=489=5.33\sigma^2 = \frac{48}{9} = 5.33

  3. Standard Deviation (σ\sigma): σ=5.33≈2.31\sigma = \sqrt{5.33} \approx 2.31

Explanation:

First, find the total frequency NN and the mean xˉ\bar{x}. Then calculate the squared deviations from the mean for each xix_i, multiply them by their respective frequencies fif_i, and find their sum. Divide this sum by NN to get the variance, and take the square root for the standard deviation.

Problem 2:

Calculate the standard deviation using the shortcut method for the following data: xx: 10, 15, 20, 25, 30 ff: 3, 2, 5, 8, 2

Solution:

Let Assumed Mean A=20A = 20. N=3+2+5+8+2=20N = 3+2+5+8+2 = 20. Calculate di=xi−20d_i = x_i - 20:

  • x=10,d=−10,fd=−30,fd2=300x=10, d=-10, f d=-30, f d^2=300
  • x=15,d=−5,fd=−10,fd2=50x=15, d=-5, f d=-10, f d^2=50
  • x=20,d=0,fd=0,fd2=0x=20, d=0, f d=0, f d^2=0
  • x=25,d=5,fd=40,fd2=200x=25, d=5, f d=40, f d^2=200
  • x=30,d=10,fd=20,fd2=200x=30, d=10, f d=20, f d^2=200

∑fidi=−30−10+0+40+20=20\sum f_i d_i = -30 - 10 + 0 + 40 + 20 = 20 ∑fidi2=300+50+0+200+200=750\sum f_i d_i^2 = 300 + 50 + 0 + 200 + 200 = 750

Using the formula: σ=∑fidi2N−(∑fidiN)2\sigma = \sqrt{\frac{\sum f_i d_i^2}{N} - \left(\frac{\sum f_i d_i}{N}\right)^2} σ=75020−(2020)2\sigma = \sqrt{\frac{750}{20} - \left(\frac{20}{20}\right)^2} σ=37.5−1=36.5≈6.04\sigma = \sqrt{37.5 - 1} = \sqrt{36.5} \approx 6.04

Explanation:

By choosing an assumed mean AA, we reduce the magnitude of the numbers. We calculate deviations did_i, then fidif_i d_i and fidi2f_i d_i^2. These values are plugged into the shortcut formula to find the standard deviation.