b. Percentiles
1) Percentile allows us to describe a given score in relation to other
scores in a distribution. It allows us to compare scores on tests that have
different means and standard deviations. A percentile is calculated as:
Number of scores less than a given score
----------------------------------------------- x 100
Total number of scores
Example: n=50 (40 had less than 90);
your score = 90
40
--- x 100 = 80
50
You achieved a higher score than 80% of
those who took the test.
2) Calculating percentile when you have a standard score: First look up
the score in the table to determine what percent of the normal curve falls
between the mean and the given score. Then, if the sign is positive, you
add the percentage to 50. If the sign is negative, you subtract the
percentage from 50.
Example: An IQ of 115 is +1z and the percentile is 34.13 + 50 = 84.13.
An IQ of 85 is -1z and the
percentile is 50 - 34.13 = 15.87
c. Standard Scores
1) Standard scores are a way of expressing a score in terms of its
relative distance from the mean. A z-score is a standard score. In
research, standard scores are used more often than percentiles.
2) Formula for a standard score:
x - M
z =
-----
s
48 - 35 13
Example: Obtain score of 48, M
= 35, SD = 5 z = -------- =
--- = 2.6
5 5
d. Transformed Standard Scores
1) Calculating z-scores results in decimals and negative numbers,
some prefer to transform them into other distributions. One distribution
that has been widely used is one with a mean of 50 and a standard deviation
of 10.
2) Transformed scores referred to as T-scores.
3) To convert a z-score to a T-score, use the following formula:
T = 10z + 50
Example: With a z-score of 2.5,
the T-score would be
T = (10)(2.5) + 50
T = 25 + 50
T = 75
e. Non-Normal Distributions
1) When a distribution does not
have relatively equal numbers on each side of the distribution but has a
large number of scores on one side, the distribution is referred to as
skewed. This disproportionate hump of scores causes a "tail" to be formed
at the opposite end of the distribution.
2) A positively skewed distribution has a tail extending on the right or
positive side of the distribution.
3) A negative distribution has a tail extending on the left or negative
side of the distribution.
4) Even a bell-shaped curve need
not be normal. The measure of relative peakedness or flatness of the curve
is called kurtosis. A narrow peaked curve is leptokurtic, and a flatter
curve is platykurtic.
f. Central Limit Theorem
1) It has been shown that when most samples are drawn from a population,
the means of these samples tend to be normally distributed. The larger the
number of samples, the more the distribution reaches the normal curve.
2) To calculate standard scores
necessary to determine position under the normal curve, need to know the
standard deviation of the distribution (z = (x - M)/s). This
new standard deviation of the means is called the: standard error of the
mean (Sx). Term error indicates the fact that due to sampling
error, each sample mean is likely to deviate somewhat from the true
population mean.
3) It has also been shown that there is a constant relationship between
the standard deviation of a distribution of sample means (standard error of
the mean), the standard deviation of the population from which the samples
were drawn, and the size of the samples. The formula for standard error of
the means is:
s
sx
= ----
__
√n
4) To summarize the central limit theorem:
As n increases:
1. The sampling distribution of the means of the samples approaches a
normal distribution.
2. The mean of these means approaches the mean of the population.
g. Confidence Intervals
1) Since means are normally distributed, we can use the standard
deviation of the distribution of means, the standard error of the mean, to
determine areas under the normal curve, and then we can determine how
confident we are that a population mean would fall within a certain
interval.
2) 95% of the distribution falls between +1.96 standard
deviations from the mean and 99% falls between +2.58 standard
deviations from the mean. Given characteristics of the normal curve, there
is a 95% probability that a given mean will fall between +1.96
standard deviations from the actual population mean and a 99% probability
that a given sample mean will fall between +2.58 standard deviations
from the actual mean.
3) To set confidence intervals, use the following formulas: (95% of the
cases is contained between +/- 1.96 SD from the mean and 99% between +/-
2.58 from the mean)
95% = M + 1.96 (sm)
99% = M + 2.58 (sm)
(sm)=standard error
Example: Sample M = 1, SD = 27, n = 81
First need to
calculate standard error of the mean: standard deviation/square root of n
Next calculate 95% CI
Mean + (1.96)
(standard error)
100 + (1.96)
(3)
100 + 5.88 =
94.12 and 105.88
27 27
sm = --- = --- = 3
___
√81 9
