Standard Distributions

Standard Distributions

In the chapter on central tendency, we defined a normal distribution as a symmetrical distribution of scores in which the mean, median, and mode are all the same. A normal distribution also has a defined standard deviation.

Together, this information helps us describe our distribution by indicating where the center of our data falls (mean) and how far spread the average point is from the center of our data (standard deviation). Now, we will talk about a special type of normal distribution called the standard normal distribution. The standard normal distribution is a normal distribution of standardized values called z-scores.

Z-Scores

A z-score is defined as:

where x is a given score in your data, μ is the mean of your data, and σ is the standard deviation of your data. As you can tell by the formula, we can convert any score into a z-score if we know the mean of the distribution, the standard deviation of the distribution, and the x value of interest. When we convert a score to a z-score, we can easily tell how far the given x value of interest is away from the mean because a z-score is in standard deviation units. That is, a z-score of 1.2 means the x value of interest is 1.2 standard deviations away from the mean.

There are some principles about z-scores that you should know. First, if a z-score is positive, this means the given x value of interest is above the mean. Alternatively, if a z-score is negative, the given x value of interest is below the mean. The mean of any normal distribution always has a z-score of 0. Following this logic, the larger the absolute value of the z-score, the farther the score is away from the mean. For example, if you took a statistics exam, and the z-score that represented your performance as 3, this means you did very well. You did 3 standard deviations above the mean, or 3 times the average spread from the mean in the distribution. If the z-score that represented your performance was negative then you quickly know that you did worse than the class average.

We could convert all of our x values into standardized scores (z-scores). If we did this, we would have what we referred to above as the standard normal distribution. The standard normal distribution always has a mean of 0 and a standard deviation of 1.

Converting our raw x scores to z-scores offers two major benefits. The first is that it allows us to compare scores across different distributions. For example, if you took a statistics exam and a chemistry exam, the distributions of the scores may be different. For the stats exam, the mean of the distribution may be 80 and the standard deviation may be 5. For the chemistry exam, the mean of the distribution may be 70 and the standard deviation may be 2. If you received a 78 on your stats exam and a 74 on your chemistry exam, which exam did you do better on? At first glance, it may appear that you did better on the stats exam (78>74), however we cannot compare these two numbers accurately because they come from distributions with different means and standard deviations. Importantly, if we convert our raw scores of 78 and 74 into z-scores first, we can then compare the z-scores to see which exam score is better. It is appropriate to make this comparison because once both exam scores are converted to z-scores, they come from the same distribution (the standard normal distribution -- which has a mean of 0 and a standard deviation of 1). Let’s convert the scores to z-scores below:

Our exam score of 78 on our statistics exam coverts to a z-score of -0.4. Quickly we can see we did worse than the average of the class. Our exam score of 74 on our chemistry exam converts to a z-score of 2. Quickly we can see we did a lot better than average on this exam. In fact, we are two standard deviations above the mean. Therefore, we can confidently say that we did better on the chemistry exam than the statistics exam, even though our raw score on our chemistry exam was less than the raw score on our statistics exam. We can only compare values if they come from the same distribution and this is why z-scores, which put values on a standard distribution, are so powerful.

The Empirical Rule

If X is a random variable and has a normal distribution with mean µ and standard deviation σ, then the Empirical Rule says the following:

About 68% of the x values lie between –1σ and +1σ of the mean µ (within one standard deviation of the mean).

About 95% of the x values lie between –2σ and +2σ of the mean µ (within two standard deviations of the mean).

About 99.7% of the x values lie between –3σ and +3σ of the mean µ (within three standard deviations of the mean). Notice that almost all the x values lie within three standard deviations of the mean.

The empirical rule is also known as the 68-95-99.7 rule.

If we think about the empirical rule in terms of the standard normal distribution, then we know that:

About 68% of the x values lie between the z-scores of -1 and +1

About 95% of the x values lie between the z-scores of -2 and +2

About 99.7% of the x values lie between the z-scores of -3 and +3.

The empirical rule tells us that if I scored 3 standard deviations greater than mean on my statistics exam (could also be written as “if I scored a z-score of 3 on my statistics exam”), then I did better than 99.7% of the people in my class. Wow, if this is the case – you must really know statistics!

The following two videos give a useful summary of the standard normal distribution.

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