- Education and Science
The Normal Distribution And Standard Normal Score
The Normal Distribution Part One
Basic definition and concepts
The normal distribution is a bell-shaped pattern of variation that was identified by the German Mathematician and astronomer Carl Gauss (1777-1855).
Gauss observed a recurring pattern of errors in repeated measurements of the same object. As an illustration take the example of a machinist who was asked to measure the diameter of 100 similar bolts – one at a time to a precision of 0,0001 inch with a micrometer. Furtherly, suppose that, unknown to the machinist all measurements were made on the same bolt. Most of the reported values would tend to cluster closely about some central value with perhaps a few observations of somewhat higher or lower values.
Many measurements such as diameter of bolts produced by the same specification have an over-all bell-shaped pattern of error. This pattern of error which was initially attributed to chance is called the “Gaussian” or “normal curve.”
As the theory of statistical inference has developed over the past 200 years, the normal distribution has played a key role. There are several reasons for this. First, many empirical data are by nature normally distributed examples are the diameters of bolts, weights of packages of foods or SAT scores.
Secondly, data that are not normally distributed can often be analyzed via the normal distribution, provided that a proper sampling technique is employed and provided the question is about a mean value or a population proportion.
Third, normal distribution has been made extremely easy to work with by virtue of a coding process that converts any particular normal distribution to a standard normal distribution. This is sufficiently common that same hand calculators have a normal distribution function programmed into their logic.
Fourth, the normal distribution can be effectively manipulated and used for making inferences.
The Standard Normal Distribution
We can evaluate areas for all normal distribution by making one simple conversion to the standard normal distribution or Z- form.
The normal or Gaussian distribution is a bell-shaped curve with its explicit shape defined by two parameters – its mean value µ and standard deviation .
The standard normal distribution is a continuous symmetrical bell-shaped distribution that has a mean of 0 and standard deviation of 1 .
These two parameters mean and standard deviation thus define the specific normal distribution.
The normal distribution is a continuous, symmetrical, bell-shaped distribution. Because it is symmetrical, half of the areas under the curve lies to the left of the central value and half lies to the right. The mean, median and mode are all the same central values, which for the standard normal distribution is zero,
Although all normal distributions are characterized by a familiar bell-shape, each one is distinguished by the location of its center and by its spread or variation about the mean.
The Standard Normal Score
The standard normal score or Z-score is a simple coding device for converting any normal distribution to a standard normal distribution. The Z-score is a measure of the number of standard deviations of distance that any point resides from the mean.
Computing for the Z-score
The formula for the standard score or Z-score is :
Z = (X - µ ) / sd
Where X = any raw score
µ = mean of the scores
sd = standard deviation
To illustrate the computation and the nature of standard scores , let us take the following scores, which are part of distribution with a mean of 60 and a standard deviation of 10.
X x Z
70 10 1.00
60 0 0.00
50 - 10 - 1.10
54 - 6 - .60
46 -14 -1.40
In the first column we have the raw scores (X). The mean is subtracted fro each of these, and this deviation from the mean or x, is divided by the standard deviation to change the deviation values into standard score values. The raw score of 60 is at the mean. There is no deviation, hence the standard score is zero. A raw score of 70 is 1 standard deviation above the mean.
70 - 60
---------------- = 1.
When we change raw scores to standard scores, we are expressing them in standard deviation units. These standard scores tell us how many standard deviation units any given raw score deviates from the mean.
Since three standard deviations on either side of the mean include practically all of the cases, it follows that the highest Z-score usually encountered is +3 and the lowest is -3.
We can describe the distribution of Z-scores by saying that they have a mean of 0 and a standard deviation of 1.
Thus anytime we see a standard score, we should be able to place exactly where an individual falls in a distribution. A student with a Z-score of 2,5 is 2,5 standard deviations above the mean on that test distribution and has a very good score. If another student got a Z-score of 0.5 it means that this student got a score which is .5 standard deviations from the mean therefore has an a performance about an average.
These standard scores are equal units of measurement and hence can be manipulated mathematically.
Sample Problem :
An apprentice plumber wants to solder a copper pipe section within an acceptable time in order to qualify as a professional pipe fitter. The requirement is that the joint be completed within one standard deviation of the professional standard time of mean = 0.7 minutes and standard deviation = 0.2 minutes. Soldering time are normally distributed.
In practice soldering, the apprentice time ranges from 0.6 minutes to 1.1 minutes with sd = 0.2 minutes. Is the apprentice ready for the test ?
Compute the number of standard deviation that 0.6 minutes and 1.1 minutes are from the mean by using the formula :
Z = (X - µ)/sd
The mean is at the center (zero point) on the Z-scale. The 0.6 time is one half unit below the mean as shown :
Z = (0.6 – 0.7)/0.2 = 0.5
So the lower limit of the welder’s time is satisfactory.
The computation for 1.1 minute time is shown below :
Z = (1.1 - 0.7)/0,2 = +2.0
Standard deviation is above the professional standard and therefore is not satisfactory.
This apprentice needs more practice to assure a time within one standard deviation of the 0.7 minute requirement.
Statistics For Business and Economics by John A. Ingram
Joseph G. Monks
Basic Statistical Methods by N. M. Downie
Robert W. Heath