Simply put, a z-score (also called a standard score) gives an idea of how far it is from the average value of a data point. More technically it is a measure of how many standard deviations below or above the given population mean a raw score.
A z-score can be placed on a normal distribution curve. The z scores go from -3 standard deviations (which would fall to the extreme left of the normal distribution curve) to +3 standard deviations (which would fall to the extreme right of the normal distribution curve). In order to use a z-score, you need to know the μ mean and the population standard deviation σ.
The basic z score formula for a sample is:
z = (x – μ) / σ
For example, let’s say you have a test score of 190. The test has a mean (μ) of 150 and a standard deviation (σ) of 25. Assuming a normal distribution, your z score would be:
z = (x – μ) / σ
= 190 – 150 / 25 = 1.6.
The z score tells you as many standard deviations from the mean your score is. In this example, your score is 1.6 standard deviations above the mean.
You may also see the z score formula shown to the left. This is exactly the same formula as z = x – μ / σ, except that x̄ (the sample mean) is used instead of μ (the population mean) and s (the sample standard deviation) is used instead of σ (the population standard deviation). Anyway, there are exactly the same steps for solving it.
Z Score Formula: Standard Error of the Mean
If you have multiple samples and want to describe the standard deviation of those sample means (the standard error), you would use this z score formula:
z = (x – μ) / (σ / √n)
This z-score will tell you that many standard errors there are between the sample mean and the population mean.
Example problem: In general, the mean height of women is 65″ with a standard deviation of 3.5″. Which is the probability of finding a random sample of 50 women with a mean height of 70″, assuming the heights are normally distributed?
z = (x – μ) / (σ / √n)
= (70 – 65) / (3.5/√50) = 5 / 0.495 = 10.1
If the key here is that we’re dealing with a sampling distribution of means, then so we know we have to include the standard error in the formula. We also know that 99% of values fall within 3 standard deviations from the mean in a normal probability distribution (see 68 95 99.7 rule). Thus, there’s less than 1% probability that whatever sample of women will have a mean height of 70″.
Confused about when to use σ and when to use σ √n? See: Sigma / sqrt (n) – why is it used?
How to calculate a Z-Score
A z-score can be easily calculated on a TI-83 calculator or in Excel. If you do not have both, however, you can calculate it by hand.
Enter your X value in the z-score equation. In this example question the X value is your SAT score, 1100.
Step 2: Put the average, μ, into the z-score equation
write the standard deviation, σ in the z-score equation.
Step 4: Find the answer using a calculator:
(1100 – 1026) / 209 = .354. That means your score was .354 devs std above average.
Step 5: (Optional) Find your z value in the z table to see what percentage of test-taker scored below you. A z-score of .354 is .1368 + .5000* = .6368 or 63.68%.
*Why add .500 to the result? The table z shown has the scores for the RIGHT of the average. So we need to add .500 for the whole LEFT area of the average. For more examples of when to add (or subtract) .500, please see several examples in: Area under a normal distribution curve.
4. 4. Z scores and standard deviations
Technically, a z-score represents the number of standard deviations from the standard value of the reference population (a population whose known values have been recorded, as in these graphs that the CDC compiles on people’s weights). For example:
A z-score of 1 is 1 above average standard deviation.
A score of 2 is 2 above average standard deviations.
A score of -1.8 is -1.8 standard deviations below average.
A z-score indicates where the score is on a normal distribution curve. A z-score of zero shows that the value is exactly the average, while a score of +3 shows that the value is much higher than the average.
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5. 5. How do you use it in real life?
You can use the z-table or the normal distribution graph to get a view of how a z-score of 2.0 means “above average”. Suppose you have the weight of a person (240 pounds) you know their z-score is 2.0. Are you aware that 2.0 is above average (because of the high positioning on the normal distribution curve), but would you like to know how much more than average this weight is?
The z-score in the middle of the curve is zero. The z-scores to the right of the average are positive and the z-scores to the left of the average are negative. If looking at the score in the z table, you can see what percentage of the population is above or below your score. The following table shows a z-score of 2.0 highlighted, showing .9772 (which converts to 97.72%). If you look at the same score (2.0) as the normal distribution curve above, you see that it corresponds to 97.72%.
It tells you that 97.72% of the population scores are below that particular score and 100% – 97.72% = 2.28% of the scores are above that score. A very simple 2.28 of the population is above this person’s weight…..probably a good indication that they need to diet!
Technology
1. Finding a Z-Score on the TI-89
The TI-89 Titanium’s Stats/List Editor contains a simple menu where you can search for a Z score in seconds. This section shows how to find the z-score for a critical value in a left tail. Your normal distribution curve is symmetrical, so this will also be the area in a right tail.
You are unsure whether your test is a left tail or a right tail? See “Left Tailed Test or Right Tailed” to help you decide.
Z-Score: Definition, Formula and Calculation
Contents (General):
What is a Z-Score?
Z Score Formulas.
How to Calculate a Z-Score.
More on Z scores and Standard Deviations.
How is it Used in Real Life?
Contents (Technology):
How To Find a Z-Score on the TI-89.
How to Find a Z-Score in Excel.
How to find a critical z-value on the TI-83.
1. What is a Z-Score?
Simply put, a z-score (also called a standard score) gives you an idea of how far from the mean a data point is. But more technically it’s a measure of how many standard deviations below or above the population mean a raw score is.
A z-score can be placed on a normal distribution curve. Z-scores range from -3 standard deviations (which would fall to the far left of the normal distribution curve) up to +3 standard deviations (which would fall to the far right of the normal distribution curve). In order to use a z-score, you need to know the mean μ and also the population standard deviation σ.
Z-scores are a way to compare results to a “normal” population. Results from tests or surveys have thousands of possible results and units; those results can often seem meaningless. For example, knowing that someone’s weight is 150 pounds might be good information, but if you want to compare it to the “average” person’s weight, looking at a vast table of data can be overwhelming (especially if some weights are recorded in kilograms). A z-score can tell you where that person’s weight is compared to the average population’s mean weight.
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2. Z Score Formulas
The Z Score Formula: One Sample
The basic z score formula for a sample is:
z = (x – μ) / σ
For example, let’s say you have a test score of 190. The test has a mean (μ) of 150 and a standard deviation (σ) of 25. Assuming a normal distribution, your z score would be:
z = (x – μ) / σ
= 190 – 150 / 25 = 1.6.
The z score tells you how many standard deviations from the mean your score is. In this example, your score is 1.6 standard deviations above the mean.
alternate-z-scoreYou may also see the z score formula shown to the left. This is exactly the same formula as z = x – μ / σ, except that x̄ (the sample mean) is used instead of μ (the population mean) and s (the sample standard deviation) is used instead of σ (the population standard deviation). However, the steps for solving it are exactly the same.
Z Score Formula: Standard Error of the Mean
When you have multiple samples and want to describe the standard deviation of those sample means (the standard error), you would use this z score formula:
z = (x – μ) / (σ / √n)
This z-score will tell you how many standard errors there are between the sample mean and the population mean.
Example problem: In general, the mean height of women is 65″ with a standard deviation of 3.5″. What is the probability of finding a random sample of 50 women with a mean height of 70″, assuming the heights are normally distributed?
z = (x – μ) / (σ / √n)
= (70 – 65) / (3.5/√50) = 5 / 0.495 = 10.1
The key here is that we’re dealing with a sampling distribution of means, so we know we have to include the standard error in the formula. We also know that 99% of values fall within 3 standard deviations from the mean in a normal probability distribution (see 68 95 99.7 rule). Therefore, there’s less than 1% probability that any sample of women will have a mean height of 70″.
Confused about when to use σ and when to use σ √n? See: Sigma / sqrt (n) — why is it used?
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3. How to Calculate a Z-Score
You can easily calculate a z-score on a TI-83 calculator or in Excel. However, if you don’t have either, you can calculate it by hand.
Example question: You take the SAT and score 1100. The mean score for the SAT is 1026 and the standard deviation is 209. How well did you score on the test compared to the average test taker?
Step 1: Write your X-value into the z-score equation. For this example question the X-value is your SAT score, 1100.
CALCULATE A Z SCORE 1
Step 2: Put the mean, μ, into the z-score equation.
CALCULATE A Z SCORE 2
Step 3: Write the standard deviation, σ into the z-score equation.
CALCULATE A Z SCORE 3
Step 4: Find the answer using a calculator:
(1100 – 1026) / 209 = .354. This means that your score was .354 std devs above the mean.
Step 5: (Optional) Look up your z-value in the z-table to see what percentage of test-takers scored below you. A z-score of .354 is .1368 + .5000* = .6368 or 63.68%.
*Why add .500 to the result? The z-table shown has scores for the RIGHT of the mean. Therefore, we have to add .500 for all of the area LEFT of the mean. For more examples of when to add (or subtract) .500, see several examples in: Area under a normal distribution curve.
Like the explanation? Check out the Practically Cheating Statistics Handbook, which has hundreds more step-by-step explanations, just like this one!
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4. Z scores and Standard Deviations
Technically, a z-score is the number of standard deviations from the mean value of the reference population (a population whose known values have been recorded, like in these charts the CDC compiles about people’s weights). For example:
A z-score of 1 is 1 standard deviation above the mean.
A score of 2 is 2 standard deviations above the mean.
A score of -1.8 is -1.8 standard deviations below the mean.
A z-score tells you where the score lies on a normal distribution curve. A z-score of zero tells you the values is exactly average while a score of +3 tells you that the value is much higher than average.
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5. How is it Used in Real Life?
You can use the z-table and the normal distribution graph to give you a visual about how a z-score of 2.0 means “higher than average”. Let’s say you have a person’s weight (240 pounds), and you know their z-score is 2.0. You know that 2.0 is above average (because of the high placement on the normal distribution curve), but you want to know how much above average is this weight?
The z-score in the center of the curve is zero. The z-scores to the right of the mean are positive and the z-scores to the left of the mean are negative. If you look up the score in the z-table, you can tell what percentage of the population is above or below your score. The table below shows a z-score of 2.0 highlighted, showing .9772 (which converts to 97.72%). If you look at the same score (2.0) of the normal distribution curve above, you’ll see it corresponds with 97.72%.
z score definition
That tells you 97.72% of the population’s scores lie below that particular score and 100% – 97.72% = 2.28% of the scores lie above that score. A mere 2.28 of the population is above this person’s weight….probably a good indication they need to go on a diet!
Technology
1. How to Find a Z-Score on the TI-89
The TI-89 Titanium’s Stats/List Editor contains a simple menu where you can look up a Z Score in seconds. This section shows you how to find the z-score for a critical value in a left tail. The normal distribution curve is symmetrical, so this will also be the area in a right tail as well.
Not sure if your test is a left tailed or right tailed? See “Left Tailed Test or Right Tailed” to help you decide.
Note that you must have the Stats/List Editor installed to be able to make a TI-89 frequency distribution using these instructions.
Z Score TI 89: Steps
Watch the video or read the steps below:
Example problem: Find the z score for α = .012 for a left-tailed test on a standard normal distribution curve.
Step 1: Press Apps, scroll to the Stats/List Editor, and press ENTER.
If you don’t see the Stats/List Editor, you can download it here. It’s an official TI app and you’ll need to transfer it to your calculator using the cable that originally came with your TI-89.
Step 2: Press F5 2 1, to get to the Inverse Normal screen.
Step 3: Enter .012 in the Area box.
Step 4: Enter 0 for the mean, μ and 1 for the standard deviation, σ.
Step 5: Press ENTER.
Step 6: Read the result: the calculator should state “Inverse = -2.25713“. This is your z score.
Tip: If you are given a mean and standard deviation, enter them in place of 0 and 1 in Step 4.
That’s how to find a z score on the TI 89!
How to find a Z-Score in Excel
Z-Score in Excel: Overview
A z-score in Excel may be rapidly calculated with a basic formula. The formula for calculating a z-score is
z=(x-μ)/σ,
where μ is the population average and σ the standard deviation of the population.
Note: When the population standard deviation is unknown or the sample size is less than 6, you should use a t-score instead of a z-score.
Z-Score in Excel: Steps
step 1: Enter the average population in an empty cell. In this example, type “469” in cell A2. Optional: Typing the word “average” as a column header in cell A1 so that you remember the value in cell A2.
Step 2: Type the population standard deviation in an empty cell. For this example, type “119” in cell B2. Optional: Enter the word “standard deviation” as the column header in cell B1 so that you remember what the value means in cell B2.
Step 3: Type the value X (in this example problem, X is your GRE score) in an empty cell. For this example, type “650” in cell C2. Optional: Type the words “X” as a column header in cell B1 so that you remember what the value in cell B2 means.
Step 4: Enter the following formula in an empty cell:
=(C2-A2)/B2
Step 5: Press “Enter”. The z-score will appear in cell D2: The z-score of 1.521008 in this sample problem indicates that your GRE score was 1.521008.
That’s it! You found a z-score in Excel.
Tip: You can use it over and over again once you have entered the formula once. Just type in a new average, a standard deviation and an X value in the corresponding boxes.
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