A Density Curve is a curve that
A mathematician named Quetelet measured all sorts of things about large samples (e.g. 5000) of people -- height, weight, eyesight -- 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 can be compared.
a. Symmetric, bell-shaped
b. 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.
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
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. Never crosses the x-axis so in theory, it is possible to have extreme observations in a normal distribution
a. First step: Draw a picture...
( i) Draw the x-axis. These are the STANDARDIZED SCORES or Z-SCORES.
( ii) Draw a normal curve above it. The area under the curve gives PERCENTAGES.
(iii) Shade in the area that you are looking for (or know). b. Solve for the area you want in terms of "left hand areas"
a. What percent of a standard normal curve is less than
1.20 standardized scores?
(first step ... draw a picture)
Table gives .8849, or 88.5%
b. What percent of a standard normal curve is greater
than 1.20 standardized scores?
(first step ... draw a picture)
100% - 88.49%, or 11.51%.
c. What percent of a standard normal curve lies between
-0.83 and 1.25?
(first step ... draw a picture)
89.44% - 20.33%, or 69.11%
d. What z-score has 3.84% area to the left of that
score?
(first step ... draw a picture)
z = -1.77
e. What z-score has 29.12% area to the right of that
score?
(first step ... draw a picture)
29.12% to the right means 100%-29.12%=70.88% to the left.
Thus z = 0.55 <--- note: this is NOT an area but a SCORE>
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=1.31, then the original score was 1.31 SD's above average; if z = -0.57, then the original score was 0.57 SD's BELOW average.
2.Formula
z = (value of interest - average)
-------------------------------------
standard deviation
3.Examples of standard units
Women's heights in the United States are normally distributed with a mean of 64 inches (about 5 feet 4 inches) and a standard deviation of 2.5 inches.
Lucy Lawless, an actress AKA Xena is 5'11" inches tall (71 inches). Her height, in standard units is:
z = (71 - 64) / 2.5 = 2.8
She is 2.8 standard units ABOVE the average height or 2.8 standard deviations above average.
4.Interpretation
Ms. Lawless is taller than 99.74% or .26% of all women are taller than she is. If you had a 1,000 selected at random, Ms. Lawless would be among the 3 tallest.
D. Why Bother with Standard Units?
Standard Units allow quick comparisons across variables with different units of measure. Z scores or standard units have no units,
everything you convert is standardized. For example, Lucy Lawless weighs 132lbs, that's a Z score of -.13 or she is lighter than (1- .4483) =
55.17% of all U.S. women. If she was as heavy as she is tall, in other words if she were to weigh more than 99.74% of all women she would
need to be:
2.8 = (value of interest - 136) /30 or 220 lbs, she'd need to gain 88lbs. Not that we particularly care about her weight, but the normal
table is a tool that can be used with a variety of datasets especially when you are working with variables with different units. It provides a single measurement scale.
1. Common sense: if the normal curve implies nonsense results (for example, that people have negative incomes, or that some women have a negative number of children), the normal curve doesn't apply and using the normal curve will give the wrong answer.
Example: in 1980, the average number of children born per woman was 1.95, with an SD of 1.91. Does the normal curve apply? Try calculating how many children a woman would have if she is 2 standard deviations BELOW the mean.
(No; the data have a long right hand tail this distribution is skewed to the right. A woman who is 2 SD below the mean has -1.87 children..)
2. Do a histogram: if the data look like a normal curve, the normal curve probably applies; otherwise, it does not.
3. Do the data fall in a 68-95-99.7% pattern?
1. If the data are normally distributed, then raw scores can be converted into standard units to find percentages; also, percentages can be converted into standard units and then converted into raw scores.
2. Examples
a. SAT math scores are normally distributed with a mean of 500 and an SD of 100. What percentile rank is a math score of 650?
(first step: draw a picture ... want an area)
(next step: convert to standard units; z = 1.50)
(final step: look up the area; rank is 93.3 percentile)
b. What SAT math score defines the top 10%?
(first step: draw a picture ... want an original score)
(next step: using the area, find the standardized score: z=1.28)
(final step: convert back from standard units; score is 628)
3. Other examples
INSTABROKER! an internet brokerage, claims that its average trade execution time is 59 seconds. It claims that the standard deviation is 15 seconds. Suppose the executions times are normally distributed. What is the chance of a trade requiring more than 90 seconds?
To answer this, you might draw a curve then fill in the information, then calculate a Z score.
z = (90 - 59) / 15 = 3.27
Your chance of having to wait more than 90 seconds is less than 1 - .9995 from the table or .0005 or .05%. In 10,000 trades, only 5 would take longer than 90 seconds.