1. Normal Distributions

2. Continuous Distribution
For a discrete distribution, for example
Binomial distribution with n=5, and p=0.4, the probability distribution is x
f(x)

3. A probability histogramx
P(x)

4. How to describe the distribution of a continuous random variable?
For continuous random variable, we also represent probabilities by areas—not by areas of rectangles, but by areas under continuous curves.
For continuous random variables, the place of histograms will be taken by continuous curves.
Imagine a histogram with narrower and narrower classes. Then we can get a curve by joining the top of the rectangles. This continuous curve is called a probability density (or probability distribution).

5. Continuous distributions
For any x, P(X=x)=0. (For a continuous distribution, the area under a point is 0.)
Can’t use P(X=x) to describe the probability distribution of X
Instead, consider P(a≤X≤b)

6. Density function A curve f(x): f(x) ≥ 0 The area under the curve is 1
P(a≤X≤b) is the area between a and b

7. P(2≤X≤4)= P(2≤X<4)= P(2<X<4)

8. Properties Of Normal Curve
Normal curves are symmetrical.
Normal curves are unimodal.
Normal curves have a bell-shaped form.
Mean, median, and mode all have the same value.
Total area = 1
Defined by mean and standard deviation
(centered at mean)

9. Percent of Values Within One Standard Deviations
68.26% of Cases
Precise relationship between the area under the normal curve and the units of the Standard Deviation.
In an ideal normal curve, the following is found if you start at the mean and go a certain number of Standard Deviations above and below it.

10. Percent of Values Within Two Standard Deviations
95.44% of Cases
Precise relationship between the area under the normal curve and the units of the Standard Deviation.
In an ideal normal curve, the following is found if you start at the mean and go a certain number of Standard Deviations above and below it.

11. Percent of Values Within Three Standard Deviations
99.72% of Cases
Precise relationship between the area under the normal curve and the units of the Standard Deviation.
In an ideal normal curve, the following is found if you start at the mean and go a certain number of Standard Deviations above and below it.

12. Percent of Values Greater than 1 Standard Deviation

13. Data in Normal Distribution

14. Standard Scores (Z-scores)
Expressed in standard deviations from mean
There are many kinds of Standard Scores. The most common standard score is the ‘z’ scores.
A ‘z’ score states the number of standard deviations by which the original score lies above or below the mean of a normal curve.

15. Commonly used probabilities and z-scores:
Middle 90%: between and
Middle 95%: between and +1.96
Middle 99%: between and
90th percentile: 1.28
95th percentile: 1.645
99th percentile: 2.33

16. Area When Score is Known
For a normal distribution with mean of 100 and standard deviation of 20, what proportion of cases fall below 80?
~16%

17. Calculator:
Normal cumulative density function: normalcdf(left bound, right bound, mean, st. dev.)
(mean and st. dev. default to 0 and 1)
To find z-scores given the area in the left tail of a normal distribution:
invNorm(area, mean, standard deviation)

18. Score When Area Is Known
For a normal distribution with mean of 100 and standard deviation of 20, find the score that separates the upper 20% of the cases from the lower 80%
Answer = 116.8

19. Ways to Assess Normality
Use graphs (dotplots, boxplots, or histograms)
Normal probability (quantile) plot

20. Normal Probability (Quartile) plots
The observation (x) is plotted against known normal z-scores
If the points on the quantile plot lie close to a straight line, then the data is normally distributed
Deviations on the quantile plot indicate nonnormal data
Points far away from the plot indicate outliers
Vertical stacks of points (repeated observations of the same number) is called granularity

21. Are these approximately normally distributed?
What is this called?
Both the histogram & boxplot are approximately symmetrical, so these data are approximately normal.
The normal probability plot is approximately linear, so these data are approximately normal.