"Population standard deviation"

Introduction to Population Standard Deviation

  • Standard deviation is a measure of the dispersion or spread of a set of data points relative to their mean. For populations, the standard deviation is denoted by σ\sigma.

Key Concepts

  • Population vs Sample: The population refers to the entire group being studied, while a sample is a subset of that population. The calculation of the standard deviation differs for populations and samples.
  • Mean (μ\mu): The average of the values in the population. It is calculated using the formula:
    μ=1n∑<em>i=1nx</em>i\mu = \frac{1}{n} \sum<em>{i=1}^{n} x</em>i where xix_i are the data points and nn is the total number of points.

Steps to Calculate Population Standard Deviation

  1. Calculate the Mean (μ\mu):

    • If the weights lost are represented as x<em>1,x</em>2,…,x<em>nx<em>1, x</em>2, … , x<em>n, the mean is: μ=x</em>1+x<em>2+…+x</em>nn\mu = \frac{x</em>1 + x<em>2 + … + x</em>n}{n}

  2. Find the Squared Deviations:

    • Calculate the squared differences from the mean for each data point:
      • (x<em>1−μ)2,(x</em>2−μ)2,…,(xn−μ)2(x<em>1 - \mu)^2, (x</em>2 - \mu)^2, …, (x_n - \mu)^2

  3. Calculate the Variance (σ2\sigma^2):

    • The population variance is the average of the squared deviations:
      σ2=1N∑<em>i=1N(x</em>i−μ)2\sigma^2 = \frac{1}{N} \sum<em>{i=1}^{N} (x</em>i - \mu)^2 where NN is the number of data points.

  4. Determine the Population Standard Deviation (σ\sigma):

    • Finally, the standard deviation is the square root of the variance:
      σ=σ2\sigma = \sqrt{\sigma^2}

Example Calculation

  • Given Weights Lost: Assuming the weights lost are 5, 4, 4, 5, 2.

  • Calculate the Mean:
    μ=5+4+4+5+25=4\mu = \frac{5 + 4 + 4 + 5 + 2}{5} = 4

  • Calculate Squared Deviations:

    • For each weights lost:
    • (5−4)2=1(5 - 4)^2 = 1
    • (4−4)2=0(4 - 4)^2 = 0
    • (4−4)2=0(4 - 4)^2 = 0
    • (5−4)2=1(5 - 4)^2 = 1
    • (2−4)2=4(2 - 4)^2 = 4

  • Variance Calculation:

    • σ2=1+0+0+1+45=65=1.2\sigma^2 = \frac{1 + 0 + 0 + 1 + 4}{5} = \frac{6}{5} = 1.2

  • Standard Deviation:

    • σ=1.2≈1.095\sigma = \sqrt{1.2} ≈ 1.095
    • Rounded to two decimal places: σ≈1.10\sigma ≈ 1.10

Conclusion

  • Population Standard Deviation Formula:
    • The general formula is:
      σ=∑<em>i=1N(x</em>i−μ)2N\sigma = \sqrt{\frac{\sum<em>{i=1}^{N} (x</em>i - \mu)^2}{N}}
  • The standard deviation provides important insights into the variability of data and is crucial in statistical analysis.