## Z Scores

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### Explanation

Introducing Z scores
We have learned that the mean of a sample tells you what the average value is and that the standard deviation tells you the average distance between the mean and all the values in the sample. It should be clear, then, that the standard deviation would be a good unit of measurement if we wanted to talk about distance from the mean.

Saying, "This value is 1 standard deviation from the mean" is the same as saying, "This value is the average distance from the mean". In this way, we could compare values from different measurements in terms of their distance from the mean. For example, you might say, "His height is one standard deviation above the mean, but his weight is one standard deviation below the mean.". Take a moment to make sure you understand why that tells us that this person is taller and thinner than average.

In statistics, the ability to talk about distances from the mean in terms of standard deviations is so useful that there is a shorthand symbol for the number of standard deviations a particular value is away from the mean. It is called a z-score.

If you know the mean and standard deviation of a sample, you can calculate the z-score for any value from that sample (or, indeed, for any new value you measure). To say, "The z-score for my IQ is 1.5." is to say that your IQ is one and a half standard deviations above the average IQ.

Here are the facts:

• A z-score is calculated for a single value and indicates the distance of that value from the mean in units of standard deviations;
• A positive z-score indicates that the value is above the mean;
• A negative z-score indicates that the value is below the mean;
• z can be any whole number or a fraction, so z = 3, z = 1.3 or z = 0.5 are all valid.
You can only use z scores if your data's distribution is symmetrical (as we saw the normal distribution is). We will see in the next topic that you can convert from z-scores to probabilities using z-tables. To do this, your data needs to be normally distributed, so it is worth checking that it is before calculating z scores.

### Exploration

Here are a few questions to get you used to working with z-scores.
Is a z score of 1.3 above or below the mean?
Is a z score of -1.3 above or below the mean?
If two values have respective z-scores of 1.2 and 2.4, which is further from the mean?
If two values have respective z-scores of -1.2 and -2.4, which is further from the mean?
If a value has a z-score of zero, what other measure is it equal to?

### Application

Here are a few examples from your own data to test your understanding.

Looking at dailyreading in the sms2 sample, which has a mean of 3.2 and a standard deviation of 1.28, a value of 4.48 taken from this data has a z score of 1
Is 4.48 above or below the sample mean?
How many standard deviations away from the mean is the value 4.48?
Another value of 2.56 taken from this data has a z score of -0.5
Is 2.56 above or below the sample mean?
How many standard deviations away from the mean is the value 2.56?

 The Normal Distribution | Probability Distributions