Fun fact: the percentage of our distribution that falls in a given area is exactly the same as the probability that any single observation will fall in that area. In other words, we know that approximately 34 percent of our data will fall between the mean and one standard deviation above the mean. We can also say that a given observation has a 34 percent chance of falling between the mean and one standard deviation above the mean.
Or, to put it another way, if you were to choose an observation at random from our distribution, there is a 34 percent chance that it would come from the area between the mean and one standard deviation above the mean. Z scores, which are sometimes called standard scores, represent the number of standard deviations a given raw score is above or below the mean.
Sometimes it's helpful to think of z scores as just another unit of measurement. If, for example, we were measuring time, we could express time in terms of seconds, minutes, hours or days. Similarly we could measure distance in terms of inches, feet, yards or miles. We might have to do a little math to convert our data from one unit of measurement to another, but the thing we are measuring remains unchanged. When we work with z scores, we're basically converting our existing data into a new unit of measurement: standard deviation units.
We can convert any raw score into z scores by using the following formula:. In other words, we just need to subtract the mean from the raw score and divide by the standard deviation. Let's go back to our distribution with a mean of 58 and a standard deviation of 5. We can convert 63 a raw score into standard deviation units z scores fairly easily:.
Just as one hour is equal to 60 minutes, a raw score of 63 in this distribution is equal to one standard deviation. The same holds true for observations below the mean:.
In this case, because our answer is negative, we know that 53 falls exactly one standard deviation below the mean. Now suppose we wanted to convert our mean 58 into a z score:.
When we convert our data into z scores, the mean will always end up being zero it is, after all, zero steps away from itself and the standard deviation will always be one. Data expressed in terms of z scores are known as the standard normal distribution, shown below in all of its glory. We simply multiply the z score by the standard deviation and add that to the mean. So if we plug the numbers from our example into the formula we get:. Once we've got our heads around the normal distribution, Kuibyshev's theorem and z scores , we can use them to determine the percentage of our data that falls in a given area of our distribution.
In order to do that, we need the cumulative z table, which I have posted on Canvas. The cumulative z table tells us what percentage of the distribution falls to the left of a given z score.
I know that the table looks pretty intimidating, so we'll spend a significant amount of time going over this in class. The normal distribution is a symmetrical, bell-shaped distribution in which the mean, median and mode are all equal. It is a central component of inferential statistics. The standard normal distribution is a normal distribution represented in z scores. The SND allows researchers to calculate the probability of randomly obtaining a score from the distribution i.
Figure 4. Proportion of a standard normal distribution SND in percentages. The probability of randomly selecting a score between Sometimes we know a z-score and want to find the corresponding raw score.
The formula for calculating a z-score in a sample into a raw score is given below:. As the formula shows, the z-score and standard deviation are multiplied together, and this figure is added to the mean. Check your answer makes sense: If we have a negative z-score the corresponding raw score should be less than the mean, and a positive z-score must correspond to a raw score higher than the mean.
S formula. S A1:A20 returns the standard deviation of those numbers. To make things easier, instead of writing the mean and SD values in the formula you could use the cell values corresponding to these values.
McLeod, S. Z-score: definition, calculation and interpretation. Simply Psychology. Toggle navigation. A z-score can be calculated from the following formula.
Here is another way to think about z-scores. A z-score is the normal random variable of a standard normal distribution. Browse Site.
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