Download Describing Distributions

Describing Distributions
Means
Standard deviation
Z scores
Normal distribution
Norms
Tracking
Means
• The arithmetic average score in a distribution
is called the mean.
Standard Deviations
• You can obtain the average squared deviation
around the mean, known as the variance.
• We need to take the square root of the variance.
The square root of the variance is the standard
deviation.
• The standard deviation is thus the square root of the
average squared deviation around the mean. Although the
standard deviation is not an average deviation, it gives a
useful approximation of how much a typical score is
above or below the average score.
• Also, we divide by N - 1 rather than N to recognize that S
of a sample is only an estimate of the variance of the
population.
Z Score
• A Z score is the difference between a score and
the mean, divided by the standard deviation.
• If a score is equal to the mean, then its Z score is
O. For example, suppose the score and the mean
are both 6; then 6 - 6 = O. Zero divided by
anything is still O.
• If the score is greater than the mean, then the Z
score is positive. If the score is less than the mean,
then the Z score is negative.
Normal distribution?
• Is it natural? Is it something intended by God?
• Yes, natural phenomena are naturally
distributed. Heights, Weights and even human
intelligence.
http://en.wikipedia.org/wiki/The_Bell_Curve
•
• In normal distribution, extremely-large values and
extremely-small values are rare and occur near the tail ends.
Most-frequent values are clustered around the mean (which
here is same as the median and mode) and fall off smoothly
in either side of it.
• In normal distribution, 68 percent of all values lie within
one standard deviation, 95.45 percent within two standard
deviations, and 99.8 within three standard deviations (called
six sigma in quality control).
• In other words, only one out of a thousand values will fall
outside of six sigma. This distribution is called 'normal' in
the sense of 'ideal' or 'standard' against which other
distributions may be compared.
Norms
• If you test all people in the population, then their
scores will be distributed naturally in a bell shape.
• The norms for a test are based on the distribution
of scores obtained by some defined sample of
individuals.
• Norms are usually assumed as a normal
distribution.
• Norms are used to give information about
performance relative to what has been observed in
a standardization sample.
• McCall’s T = 10Z + 50
• Qyartiles: divide the frequency distribution into
equal fourths (25% intervals).
• Deciles divide: the frequency distribution into
equal 10ths (10% intervals).
• Stanine system: ranges from 1 to 9 with a mean of
5 and a SD of approximately 2.
• IQ: a mean of 100 and SD of 15 used for most of
intelligent tests
Tracking
• Children who are small as infants often remain
small and continue to grow at a slower pace
than do others.
• This tendency to stay at about the same level
relative to one’s peers is known as tracking.
Height and weight are good examples of
physical characteristics that track. Figures
An example of a child going off track.
Questions
• My Z score is 0.8. Am I above average? Can you
estimate my percentile?
• Crystal got 89 (Z score 1.1) in her class exam and
Leo got 93( Z score .07) in his class exam. (They
are in different classes) Who did better?
• If you test all the people in a population, their
scores will be naturally distributed in a
symmetrical bell shape. It is called ____________.
• If my IQ is 130, my percentile rank is _ and my Z
score ______.
• Why do we need norms?