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# Z-Score

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Have you ever read a research study and wondered how the researchers draw conclusions from the data they collect?

In research, scientists use statistics to analyze the data they collect and figure out what it means. There are a lot of ways to organize and analyze data, but one common way is converting raw scores into z-scores.

• What is a z-score?
• How do you calculate a z-score?
• What does a positive or negative z-score mean?
• How do you use a z-score table?
• How to calculate a p-value from a z-score?

## Z-Score in Psychology

Many psychological studies use statistics to analyze and better understand the data collected from the studies. Statistics turn a participant's results in a study into a form that allows the researcher to compare it with all the other participants. Organizing and analyzing the data from a study helps researchers draw meaningful conclusions. Without statistics, it would be really hard to understand what the results of a study mean by itself, and compared to other studies.

A z-score is a statistical value that helps us compare a piece of data to all of the other data in a study. Raw scores are the actual results of the study before performing any statistical analysis. Converting raw scores to z-scores helps us figure out how one participant's results compare to the rest of the results.

One way to test the efficacy of a vaccine is to compare the results of a vaccine trial to the efficacy of vaccines used in the past. Comparing the results of a new vaccine to the efficacy of an old vaccine requires z-scores!

Replication of research is super important in psychology. Conducting research on something one time is not enough; the research needs to be repeated many times with different participants of different ages in different cultures. The z-score offers researchers a way to compare the data from their study to the data from other studies.

Maybe you want to replicate a study about whether studying all night before a test helps you get a better score. After you implement your study and collect your data, how are you going to compare the results of your study with older material? You will need to convert your results to z-scores!

A z-score is a statistical measure that tells you how many standard deviations a specific score lies above or below the mean.

That definition sounds really technical, right? It's actually pretty simple. The mean is the average of all the results from the study. In a normal distribution of scores, the mean falls directly in the middle. Standard deviation (SD) is about how far the rest of the scores are away from the mean: how far the scores deviate from the mean. If the SD = 2, you know that the scores fall pretty close to the mean.

In the image of a normal distribution below, check out the z-score values near the bottom, right above the t-scores.

Normal distribution chart, Wikimedia Commons

## How to Calculate a Z-Score

Let's take a look at an example of a situation when calculating a z-score would come in handy.

A psychology student named David just took his psychology 101 exam and scored 90/100. Among David's class of 200 students, the average test score was 75 points, with a standard deviation of 9. David would like to know how well he did on the exam compared to his peers. We need to calculate David's z-score to find the answer to that question.

What do we know? Do we have all of the data we need to calculate a z-score? We need a raw score, the mean, and the standard deviation. All three are present in our example!

### Z-Score Formula and Calculation

We can calculate David's z-score using the formula below.

Z = (X - μ) / σ

where, X = David's score, μ = the mean, and σ = the standard deviation.

Now let's calculate!

z = (David's score - the mean) / the standard deviation

z = (90 - 75) / 9

Using the order of operations, perform the function inside the parentheses first.

90 - 75 = 15

Then, you can perform the division.

15 / 9 = 1.67 (rounded to the nearest hundredth)

z = 1.67

David's z-score is z = 1.67.

## Interpreting Z-Score

Great! So what does the number above, i.e., David's z-score, actually mean? Did he perform better than most of his class or worse? How do we interpret his z-score?

### Positive and Negative Z-Score

Z-scores can be either positive or negative: z = 1.67, or z = –1.67. Does it matter whether the z-score is positive or negative? Absolutely! If you look inside a statistics textbook, you'll find two types of z-score charts: ones with positive values, and ones with negative values. Check out that image of a normal distribution again. You'll see that half of the z-scores are positive and half are negative. What else do you notice?

Z-scores that fall on the right side of a normal distribution or above the mean are positive. David's z-score is positive. Just knowing that his score is positive tells us that he did as well or better than the rest of his classmates. What if it was negative? Well, we would know automatically that he only did as well as or worse than the rest of his classmates. We can know that just by looking to see if his score is positive or negative!

### P-Values and Z-Score

How do we take David's z-score and use it to figure out how well he did on the test compared to his classmates? There is one other score that we need, and it is called a p-value. When you see "p", think probability. How probable is it that David got a better or worse score on the test than the rest of his classmates?

Z-scores are great for making it easier for researchers to obtain a p-value: the probability that the mean is higher than or equal to a specific score. A p-value based on David's z-score will tell us how likely it is that David's score is better than the rest of the scores in his class. It tells us more about David's raw score than the z-score does alone. We already know that David's score is better than most of his class on average: But how much better is it?

If most of David's class scored pretty well, the fact that David also scored well isn't that impressive. What if his classmates got a lot of different scores with a wide range? That would make David's higher score much more impressive compared to his classmates! So, in order to figure out how well David did on the test compared to his class, we need the p-value for his z-score.

### How to Use a Z-Score Table

Figuring out a p-value is tricky, so researchers have created handy charts that help you quickly figure out p-values! One is for negative z-scores, and the other is for positive z-scores.

Positive Z-score table, StudySmarter Original

Negative z-score table, StudySmarter Original

Using the z-score table is pretty easy. David's z-score = 1.67. We need to know his z-score in order to read the z-table. Take a look at the z-tables above. On the far left column (y-axis), there is a list of numbers ranging from 0.0 to 3.4 (positive and negative), while on the row across the top (x-axis), there is a list of decimals ranging from 0.00 to 0.09.

David's z-score = 1.67. Look for 1.6 on the y-axis (left column) and .07 on the x-axis (top row). Follow the chart to the place where the 1.6 on the left meets the .07 column, and you will find the value 0.9525. Make sure you are using the positive z-score table and not the negative one!

1.6 (y-axis) + .07 (x-axis) = 1.67

That's it! You found the p-value. p = 0.9525.

No calculations are required to use the table, so it is quick and easy. What do we do with this p-value now? If we multiply the p-value by 100, that will tell us how well David scored on the test compared to the rest of his class. Remember, p = probability. Using the p-value will tell us what percentage of people scored lower than David.

p-value = 0.95 x 100 = 95 percent.

95 percent of David's peers scored lower than him on the psychology exam, which means that only 5 percent of his peers scored higher than him. David did pretty well on his exam compared to the rest of his class! You just learned how to calculate a z-score, find a p-value using the z-score, and turn the p-value into a percentage. Great job!

## Z-Score - Key takeaways

• A z-score is a statistical measure that tells you how many standard deviations a specific score lies above or below the mean.
• We need a raw score, the mean, and the standard deviation to calculate a z-score.
• The formula for a z-score is Z = (X - μ) / σ .
• Negative z-scores correspond to raw scores that lie below the mean while positive z-scores correspond to raw scores that lie above the mean.
• The p-value is the probability that the mean is higher than or equal to a specific score.
• Z-scores allow us to use z-tables to find the p-value.
• z-score = 1.67. Look for 1.6 on the y-axis (left column) and .07 on the x-axis (top row). Follow the chart to the place where the 1.6 on the left meets the .07 column, and you will find the value 0.9525. Rounded to the nearest hundredth, the p-value is 0.95.
• P-values can be converted to percentages: p-value = 0.95 x 100 = 95 percent.

To find a z-score, you will need to use the formula z=(x-μ)/σ.

A z-score is a statistical measure that indicates the number of standard deviations a given value lies above or below the mean.

Yes, a z-score can be negative.

No, standard deviation is a value that measures the distance a group of values lies relative to the mean, and a z-score indicates the number of standard deviations a given value lies above or below the mean.

A negative z-score means that a given value lies below the mean.

## Final Z-Score Quiz

Question

True or False: Raw scores are the actual results of a study before performing any statistical analysis.

True

Show question

Question

True or False: Researchers convert z-scores into raw scores.

False

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Question

Using the given information, find the z-score. ***Round to TWO decimal places.

X: 99

Mean: 94

Standard Deviation: 6

0.83

Show question

Question

Using the given information, find the z-score. ***Round to TWO decimal places.

X: 118

Mean: 102

Standard Deviation: 10

1.60

Show question

Question

Using the given information, find the z-score. ***Round to TWO decimal places.

X: 234

Mean: 230

Standard Deviation: 8

0.50

Show question

Question

True or False: Z-scores can never be positive.

False

Show question

Question

Fill in the blank: Z-scores that fall on the right side of a normal distribution or above the mean are ________.

positive

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Question

True or False: P-values can be converted into percentages.

True

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Question

Fill in the blank: Negative z-scores correspond to raw scores that lie _______ the mean.

below

Show question

Question

True or False: Positive z-scores correspond to raw scores that lie above the mean.

True

Show question

Question

Use the following p-value and convert it into a percentage.

P-Value = 0.95

95%

Show question

Question

Use the following p-value and convert it into a percentage.

P-Value = 0.55

55%

Show question

60%

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