Standard deviation is a statistic that characterizes a distribution of scores. It increases in direct proportion as the scores spread out more widely, the larger the standard deviation, the wide the spread of scores.
The meaning of standard deviation is best defined by normal distribution of scores. The normal distribution is illustrated by the normal curve. Normal curve is a symmmetrical curve having a bell-like shape. The total area under the normal curve represents all the scores in a normal distribution. In such a curve, the mean, the median, and the mode are identical, so the mean falls at the exact center of the curve. The curve has no boundaries in either direction, for the curve never touches the baseline no matter how far it is extended. The curve is a curve of probability, not certainty.
Practical Applications of the Normal Curve
In the field of educational reserach, there are a number of practical applications of the normal curve, among which are:
1. To calculate the percentile rank of scores in a normal distribution.
2. To normalize a frequency distribution, which is an importangt process in standardizing a psychological test or inventory.
3. To test the significance of observed measures in experiments, relating them to the chance fluctuations or errors that are inherent in the process of sampling and generalizing about populations from which the samples are drawn.
Standard Scores
An important feasture of a normal distribution is that its mathematical equation is such that one can determine the are under the curve between any two points on the horizontal scale if he knows its mean and its standard deviation. Researchers are often interested in seeing how are person's score compare with another's. To determine this, researchers convert raw scores to derived scores such as standard scores.
Standard scores use a common scale to indicate how an individual compares to other individuals in a group. These scores are particularly helfful in comparing an individual's relative position on different instruments.
Two Most frequency used standard scores
Z - scores. These standard scores tell how far a raw score is from the mean in standard deviation units. The formula is
Z = x - mean/ s
where: X = any raw score, x^ = mean, s = standard deviation of the score distribution
t - scores . These are z scores that are expressed in another way. The formula is:
t = 50 + (x - mean)/s or 50 + 10z
Examples:
1. A Senior student received a grade of 84 on the Final examination in science for which the mean grade was 76 and the standard deviation was 10. On the final examination in mathematiocs for which the mean grade was 82 and the standard deviation was 16, he received a grade of 90. In which subject was his relative standing higher?
2. A teacher wanted to get student's equally weighted mean achievement on Algebra and English test. The data are shown below:
Course Test score mean highest possible Standard deviation
Algebra 50 57 70 5
English 94 120 200 20
NORMAL DISTRIBUTION
The normal probability distribution is a continuous
distribution which is regarded by many as the
most significant probability distribution in the theory of statistics,
particularly in the field of statistical inference. It is graphically represented by a
symmetrical bell-shaped curve better known as the normal curve.
1.
The mean, median and mode have the same value,
and therefore are plotted on the same point along the horizontal axis.
2.
The curve is symmetric about the vertical line
which contains the mean.
3.
The curve is asymptotic to the horizontal axis;
that is the curve extends indefinitely in both directions.
4.
The total area under the normal curve is equal
to 1.
1.
The IQ score of a large group of students are
approximately normally distributed with a mean of 100 and a standard deviation
of 15. What is the probability that a
randomly chosen student from this group will have an IQ score:
a)
Above 120
b)
Below 128
c)
Below 93
d)
Between 85 and 110
e)
Between 115 and 125
2.
A Computer instructor constructed a learning
module aimed at familiarizing new students basic EDP concepts. Past experience has shown that the length of
time required by new students to complete the module is normally distributed
with a mean of 250 hours and standard deviation of 50 hours. What is the probability that a random selected
new students will require
a)
More than 350 hours to complete the module
b)
Between 250 and 300 hours
c)
Less than 200 hours
d)
Between 270 and 310
e)
Above 310 hours
3.
Consider a normal distribution with a mean of
500 and a standard deviation of 50. Find
the probability
a)
More than 400
b)
Between 300 and 450
c)
Less than 360
d)
Between 330 and 430
e)
Greater than 440
4.
Given a normal distribution with a mean of 180
and a standard deviation of 20. Find the
probability of
a)
Area above 230
b)
The area below 215
c)
The area above 170
d)
The area between 160 and 200
e)
The area between 185 and 225
5.
A normal distribution has a mean of 82 and a
standard deviation of 5. Find the
probability
a)
Area above 90
b)
Area below 78
c)
Area below 93
d)
Area between 75 and 80
e)
Area between 85 and 95
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