Normal Distribution

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This post covers Normal Distribution.

Ztable: https://www.mathsisfun.com/data/standard-normal-distribution-table.html

Ztable: Z

  • Different ways that data can be distributed

  • Examples of Normal Distribution

    • Heights of people
    • size of things produced by machines
    • errors in measurements
    • blood pressure
    • marks on a test

Normal Distribution

  • mean = median = mode
  • symmetry about the center
  • 50% of values less than the mean and 50% greater than the mean

Standard Deviation

  • measure of how spread out numbers are
  • square root of the Variance
  • Variance is average of the squared differences from the Mean

  • Example
  • Heights: 600mm, 470mm, 170mm, 430mm and 300mm
  • Compute Mean, the Variance, and the Standard Deviation
  • Mean
    • 394
  • Variance
    • Each Dog’s Difference from the mean
    • 21704
  • Standard Deviation
    • 147.32
    • SD is useful since we can show which heights are within one Standard Deviation (147) of the mean (394 mm)
    • Using Standard Deviation, we have a standard way of knowing what is normal and what is extra large, or extra small
  • Correction for Sample Data
    • If the data is population, then variance is average of squared differences
    • If the data is sample from a bigger population, we divide by N-1 for calculating variance
    • Sample Variance: 27130
    • Sample Standard Deviation: 165
  • Example
    • 95% of students are between 1.1m and 1.7m tall. Assume data is normally distributed, compute mean and standard deviation
    • Mean is halfway between 1.1m and 1.7m
      • Mean = (1.1m + 1.7m) / 2 = 1.4m
    • 95% is 2 standard deviations either side of the mean (a total of 4 standard deviations) so:
      • $1~SD = \frac{1.7m-1.1m}{4}=0.15$

It is good to know the standard deviation, because we can say that any value is:

  • likely to be within 1 standard deviation (68 out of 100 should be)
  • very likely to be within 2 standard deviations (95 out of 100 should be)
  • almost certainly within 3 standard deviations (997 out of 1000 should be)

Standard Scores

  • Example
    • One student is 1.85m tall
    • 1.85m is 3 standard deviations from the mean of 1.4
      • $ \frac{1.85 - 1.4}{.15} = \frac{.45}{.15}=3 $
    • Thus, z-score is 3.0
  • Example
    • 26, 33, 65, 28, 34, 55, 25, 44, 50, 36, 26, 37, 43, 62, 35, 38, 45, 32, 28, 34
    • Mean is 38.8 minutes, and the Standard Deviation is 11.4 minutes
  • Why Standardize?
    • Marks out of 60
      • 20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17
    • Mean = 23 and Standard Deviation = 6.6
    • -0.45, -1.21, 0.45, 1.36, -0.76, 0.76, 1.82, -1.36, 0.45, -0.15, -0.91
      • Only two lower than one SD

  • Your score in a recent test was 0.5 standard deviations above the average, how many people scored lower than you did?
    • Between 0 and 0.5 is 19.1%
    • Less than 0 is 50% (left half of the curve)
    • So the total less than you is:
      • 50% + 19.1% = 69.1%

References

  • https://www.mathsisfun.com/data/standard-normal-distribution.html
  • https://www.statisticshowto.com/probability-and-statistics/normal-distributions/