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WIKI: THE NORMAL DISTRIBUTION
1 What is the normal distribution?
The normal distribution is a probability distribution with the mean as the highest value. It is denser in parts close to the mean, and becomes less dense as the values move farther from the mean.
2 Why do I need to learn the normal distribution?
Although it may be true that it is almost impossible to find a data set that shows perfectly normal distribution, there are many real life data that approximate the normal distribution closely enough. These data sets can be analyzed with a certain degree of accuracy if they are assumed to be normal.
Data such as height, weight, and IQ are just some examples of real life data that follow a normal distribution.
Moreover, a sampling distribution becomes closer and closer to a normal distribution, as the number of samples increases. This is the essence of the Central Limit Theorem. This sampling distribution is created by getting different samples from a population, and calculating a statistic such as the mean for each sample. The distribution of the means is referred to as a sampling distribution.
3 What does the normal distribution look like?
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4 How do I construct and analyze a normal distribution?
Before we can discuss a normal distribution, we first need to define some terms.
An experiment is an action such as tossing a coin, rolling a spinner, or drawing a card from a deck whose results have an element of uncertainty. This is why experiment outcomes use the concept of probability.
An outcome is the result of an experiment.
Probability refers to the chance that an event will occur. In the context of an experiment, we can calculate the probability of a given outcome using the following formula.
P(event) = (# of favorable outcomes) / (total number of possible outcomes)
Example 1 : Suppose a coin is tossed (experiment). The result could either be a head or a tail (outcomes).
P(head) = 1/2 = 0.5 = 50%
There is a 50% chance that the coin will turn up a head.
A random variable is a variable that can take on numerical values assigned to outcomes of an experiment.
Example 2 : Suppose 2 coins are tossed. Let X = the number of heads that turn up. X is a random variable, which can be assigned a value of 0, 1, or 2.
When 2 coins are tossed, the set of possible outcomes is {HH, HT, TH, TT}. This set of all possible outcomes is called the sample space of the experiment.
Each outcome for the random variable X corresponds to a probability, denoted P(X = x). We can use the sample space to determine each probability.
P(X= 0) = probability that there is no head = P(TT) = 1/ P(X = 1) = probability that there is 1 head = P(HT or TH) = 2/4 = 1/ P(X = 2) = probability that there are 2 heads = P(HH) = 1/
We can organize this information into a probability distribution table, as shown below.
To convert the values to a normal score or a z score, we use the following formula: z = (x 灰 ) / 烠
Example 3 : The scores of 20 students in a 30 point test are recorded. The scores have a mean of 23, and a standard deviation of 1.5. What z score corresponds to a raw score of 20?
z = (20 23)/1.5 = 2
This means that a score of 20 is 2 standard deviations away from the mean. Since the z score is negative, then the observed score is 2 standard deviations less than the mean.
Another number that statisticians are interested in is the area under the curve , bounded by the x axis and a given z score. To get the area under a curve corresponding to a z score, we refer to a statistical tool called the Areas Under the Standard Normal Curve Table.
To read the table, refer to the first column for the whole number and tenths place. Refer to the first column for the hundredths place of the decimal. For a z score of 2, or 2.00, we can locate the area in the following way. The table shows that the area under the standard normal curve that corresponds to a z score of 2 is 0.4772.
