A mathematician named Quetelet measured all sorts of things using large samples (e.g. 5000) of people -- height, eyesight, hearing ability -- and found a regular pattern. Most people were average and smaller numbers were below average and above average. The distribution of traits had a "bell shape." This curve is an ideal to which other distributions, such as histograms, can be compared.
a. Symmetric, bell-shapedb. Mean 0, SD 1
c. The median is where 50% (half) of the observations are on either side. In this distribution, the mean is equal to the median. Values above the average are positive, values below the average are negative.
d. Area under the curve is equal to 1 or (100% when expressed as a proportion. Area under the curve represent proportions of the observations.
e. 68%-95%-almost 100% rule (see p. 64)
About 68% fall within plus or minus 1 SD of the mean
About 95% fall within plus or minus 2 SD of the mean
Nearly 100% (99.7%) fall within plus or minus 3 SD
f. The curve never crosses the horizontal axis, it gets very close at the extremes though.
A score z is in STANDARD UNITS if tells how many SD's the original score is above or below the average. For example, if z=1.3, then the original score was 1.3 SD's above average; if z = -0.57, then the original score was 0.57 SD's BELOW average.
z = (observation of interest - average)
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standard deviation
Women's heights in the United States are normally distributed with a mean of 63.5 inches (about 5 feet 4 inches) and a standard deviation of 2.5 inches.
Julia Roberts, an actress is 5'10" inches tall (70 inches). Her height, in standard units is:
z = (70 - 63.5) / 2.5 = 2.60
She is 2.60 standard units ABOVE the average height.
Julia Roberts is taller than 99.5% of all women in the US.
How do we know this? See Table A on page A-105 in the very back of your book.
Look up 2.60. Ignore the 1.36 next to it and focus on the 99.07. The interpretation is as follows, 99.07% of all women are between +2.60 and -2.60.
By symmetry, 49.54% (or 1/2 of 99.07) are between 0 and +2.60 (where Julia stands) and we know that 50% are less than zero (remember how much total area is under the curve). 49.54+50 = 99.54%
At the other extreme, a different woman is 4'11" tall (59 inches). Her height in standard units is:
z = (59 - 63.5) / 2.5 = -1.80
She is 1.80 standard units BELOW average height. From the table on A 105, 92.81% of all women are between +1.80 and -1.80.
In other words, she is taller than only 3.6% of all women Or is shorter than 96.4% of all women.
Suppose I tell you that a woman's standardized height is -1.56. How tall is she?
actual height = (-1.56*2.5) + 63.5 = 59.6
She is 59.6 inches tall or about 4'11".
Standard Units allows quick comparison across variables with different units of measure. For example, Julia Roberts weighs in at 120 lbs...that's a Z of -1.2...how much would she need to weigh if she were FATTER than 99% of all women... We're assuming an average weight of 142lbs and an SD of 18 lbs. About 189lbs....
1. The highest point of the curve is the mean.
2. The median is the same as the mean.
3. The normal curve is symmetric about its middle.
4. As you move away from the middle in either direction, the height decreases in such a way that the curve has a bell-shaped appearance
5. The total area under the curve is 100%
6. 68% of the area will be within +1 and -1 SD from the mean. 95% will fall within +2 and -2 SD from the mean. 99.7% will fall within +3 or -3 SD.
7. The curve never actually crosses the horizontal axis. It gets close (and the area under it gets very small) but it never crosses.