Download 1.) Describe the properties of a normal distribution. Why are there

1.) Describe the properties of a normal distribution. Why are there infinite possible normal distributions? Why should one assume
that sample data represents a population distribution?
The normal distribution is a family of probability distributions of the same general form, but differing in their position
and parameters. The parameters of the normal distribution are (a) The mean and (b) The standard deviation.
The standard normal distribution is the normal distribution with a mean of 0 and a variance of 1.
The main characteristics of the normal distribution are:
(1) The distribution is bell shaped
(2) The normal probability density function is symmetric about its mean value.
(3) The mean, median and mode of the distribution are all equal.
(4) The inflection points (The points where the curve changes its concavity) of the curve occur at one standard deviation
away from the mean.
There are many normal distributions because there is one normal distribution for each mean and standard deviation.
(Every situation is likely to have different mean and/or different standard deviation)
To enable statistical analysis, some assumption has to be made regarding the parent distribution. It is customary to
assume that the population from which the sample is drawn has a normal distribution, so that formulas can be used in
the analysis.
2.) (a) What is the purpose of developing a frequency distribution? (b) What is the relationship between the relative frequency and
the cumulative frequency? (c) Could a frequency distribution contain qualitative data? Why?
(a) A frequency distribution is developed so that a large (unmanageable) data can be broken down into small
manageable units. In creating a frequency distribution, we distribute the data into bins of suitable size. For example,
imagine we had the class marks of 200 students in a Math quiz. This data is large for handling. The marks scored (where
the maximum is 25) can be distributed into bins, say 1-5, 6-10, 11-15, 16-20 and 21-25. The distribution then shows the
number of students whose scores fell within each bin. Simple procedures are available to calculate the descriptive
statistics (measures of central tendency and dispersion) for such frequency distributions.
(b) Relative frequency for each bin or interval is the number of observations in that interval divided by the total number
of observations. For example, if 10 students have scored marks in the interval 1-5, the relative frequency for this class
will be 10/200 = 0.05
Cumulative frequency is the “running sum” of the frequencies. For example if the frequencies for classes 1-5, 6-10, 11-15
are respectively 10, 22 and 58, the cumulative frequencies for these classes will be respectively 10, 10 + 22 = 32 and
32 + 58 = 90. This is the relationship between frequency and cumulative frequency. Similar relationship holds between
relative frequency and cumulative relative frequency. In any distribution, the total cumulative relative frequency is 1.
(c) Yes, qualitative data can be represented as a frequency distribution. For example on a cold day, you can count how
many students are wearing red sweaters, blue sweaters, green sweaters etc and present this information as a frequency
distribution.