3P15, Lecture 4, Jan 29
Haan: An Introduction to Statistics for Canadian Social Scientists
Chapter 7: Standard Deviations, Standard Scores, and the Normal Curve
Textbook corrections for OLD text
• P.62., where the formula for the z-score appears, and then a calculation. The
denominator got lost part-way through the calculation. i.e., It should be (20 – 32.3) / 10.2
= -12.3/10.2 = -1.21. [This is indeed the number that appears below].
• The last line of that paragraph reads, “You’ll need to subtract the value for 1.21 from 1”.
Scratch that out.
• You just have to add the two B values (see next slide) to get the percent between 20 and
40. [This is what was done in the text].
• For clarity, I used the values that are used in the text above. But, these don’t match the
values from table 7.1 or Appendix A. In 7.1, they are 0.385 and 0.288. In Appendix A,
they are .3869 and .2734. So the percentages that follow are also (slightly) incorrect.
Review: Asymptotic Normality
Samples=10000, Population=Normal, N=200
Lines drawn at -2sd -1sd mean +1sd +2sd
-4 -2 0 2 4
• Distribution of a variable will approach/achieve normality with an infinite number of
• The larger the sample, the better
Some terms used to discuss variable distributions
• Measures of central tendency (mean, median, and mode)
• Measures of Variation (variance, standard deviation, range, outliers, etc.) • Measures of normality (skewness and kurtosis)
• Area under the curve
• z-scores and probabilities
Area under the Curve
• Refers to the area that lies between the normal curve and the baseline
• In a completely normal distribution, 100% of all cases should be ‘below’ the normal curve
Some Distributional Characteristics of the Normal Distribution
• Proportion of observations between each red line
o +/- 1 s.d of the mean = 68.26% (68%)
o +/- 2 s.d of the mean = 95.44% (95%)
o +/- 3 s.d of the mean = 99.74% (99%)
Okay to round as we can then have nice, easy numbers to think about
Normal Curve with Standard Deviations
Standard deviations, revisited
• By knowing (or being able to calculate) the standard deviation, it is possible to estimate
what percentage of observations lie at any given point of interest.
• This is because the standard deviation is a unit of measurement.
• The standard deviation is useful for ranking some observations on the normal curve.
• What happens if you want to k