what are true of z scores which are equal to the mean?

Contents (Full general):

  1. What is a Z-Score?
  2. Z Score Formulas.
  3. How to Calculate a Z-Score.
  4. More on Z scores and Standard Deviations.
  5. How is it Used in Real Life?

Contents (Engineering science):

  1. How To Observe a Z-Score on the TI-89.
  2. How to Find a Z-Score in Excel.
  3. How to find a critical z-value on the TI-83.
  4. Z Scores in SPSS (New Window)

one. What is a Z-Score?

Watch the video to acquire what a z-score is.

Tin can't see the video? Click hither.

Simply put, a z-score (also called a standard score) gives y'all an idea of how far from the mean a data point is. But more than technically it'south a measure of how many standard deviations below or above the population mean a raw score is.

A z-score tin can be placed on a normal distribution bend. Z-scores range from -3 standard deviations (which would fall to the far left of the normal distribution bend) 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 demand to know the hateful μ and too the population standard deviation σ.

Z-scores are a way to compare results to a "normal" population. Results from tests or surveys take thousands of possible results and units; those results can ofttimes seem meaningless. For example, knowing that someone'due south 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 tin tell yous where that person'due south weight is compared to the average population'due south mean weight.

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2. Z Score Formulas

The Z Score Formula: I Sample

The bones z score formula for a sample is:

z = (10 – μ) / σ

For example, permit's say you have a examination score of 190. The test has a hateful (μ) of 150 and a standard divergence (σ) of 25. Bold a normal distribution, your z score would be:

  • z = (x – μ) / σ
  • = (190 – 150) / 25 = ane.6.

The z score tells y'all how many standard deviations from the mean your score is. In this case, your score is i.6 standard deviations above the hateful.

alternate-z-scoreY'all may also meet the z score formula shown to the left. This is exactly the same formula equally z = x – μ / σ, except that x̄ (the sample mean) is used instead of μ (the population hateful) and southward (the sample standard deviation) is used instead of σ (the population standard departure). Withal, the steps for solving it are exactly the same.

Z Score Formula: Standard Fault of the Mean

When y'all have multiple samples and want to describe the standard deviation of those sample means (the standard error), y'all would use this z score formula:
z = (ten – μ) / (σ / √n)
This z-score will tell you how many standard errors in that location are between the sample mean and the population mean.

Case problem: In full general, the mean superlative of women is 65″ with a standard deviation of 3.5″. What is the probability of finding a random sample of fifty women with a mean height of 70″, assuming the heights are normally distributed?

  • z = (x – μ) / (σ / √north)
  • = (70 – 65) / (3.5/√50) = five / 0.495 = 10.1

The fundamental here is that we're dealing with a sampling distribution of ways, so we know we accept to include the standard mistake in the formula. We also know that 99% of values fall within 3 standard deviations from the hateful in a normal probability distribution (see 68 95 99.7 rule). Therefore, in that location's less than one% probability that any sample of women will accept a mean acme of lxx″.

Confused virtually when to use σ and when to utilise σ √northward? See: Sigma / sqrt (northward) — why is information technology used?
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3. How to Calculate a Z-Score

Y'all can easily calculate a z-score on a TI-83 estimator or in Excel. Nevertheless, if you lot don't have either, you tin can calculate information technology by hand.

How to summate a z score in statistics

Example question: You accept the SAT and score 1100. The mean score for the Saturday is 1026 and the standard departure is 209. How well did you score on the test compared to the boilerplate test taker?

Step i: Write your X-value into the z-score equation. For this example question the X-value is your Sabbatum score, 1100.
CALCULATE A Z SCORE 1

Stride 2: Put the hateful, μ, into the z-score equation.
CALCULATE A Z SCORE 2

Step three: Write the standard divergence, σ into the z-score equation.
CALCULATE A Z SCORE 3

Step 4: Find the respond using a estimator:
(1100 – 1026) / 209 = .354. This means that your score was .354 std devs to a higher place the mean.

Step 5: (Optional) Look up your z-value in the z-table to see what percentage of test-takers scored beneath 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 hateful. Therefore, we have to add together .500 for all of the expanse LEFT of the mean. For more examples of when to add (or subtract) .500, run into several examples in: Area under a normal distribution curve.

Like the explanation? Check out the Practically Cheating Statistics Handbook, which has hundreds more stride-by-footstep explanations, but 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 hateful.
  • A score of two is 2 standard deviations above the mean.
  • A score of -ane.8 is -i.8 standard deviations beneath the hateful.

A z-score tells you lot 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 yous that the value is much higher than average.

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5. How is it Used in Real Life?

Y'all can employ the z-table and the normal distribution graph to requite you a visual about how a z-score of ii.0 ways "higher than boilerplate". Let's say you take a person'south weight (240 pounds), and you know their z-score is 2.0. Yous know that ii.0 is above average (because of the loftier placement on the normal distribution curve), merely you want to know how much above average is this weight?

The z-score in the center of the curve is cypher. The z-scores to the correct of the mean are positive and the z-scores to the left of the mean are negative. If you expect up the score in the z-table, you can tell what pct of the population is in a higher place or below your score. The tabular array below shows a z-score of 2.0 highlighted, showing .9772 (which converts to 97.72%). If you await at the aforementioned score (2.0) of the normal distribution bend above, you'll see it corresponds with 97.72%.

z score definition
That tells y'all 97.72% of the population's scores lie below that item score and 100% – 97.72% = ii.28% of the scores lie above that score. A mere 2.28 of the population is to a higher place this person'south weight….probably a good indication they demand to go on a diet!

Technology

1. How to Detect a Z-Score on the TI-89

The TI-89 Titanium's Stats/List Editor contains a simple menu where you can await up a Z Score in seconds. This department shows you how to find the z-score for a critical value in a left tail. The normal distribution curve is symmetrical, then this will also be the area in a right tail as well.

Not certain if your test is a left tailed or correct tailed? See "Left Tailed Exam or Right Tailed" to help you make up one's mind.

Annotation that yous must take the Stats/Listing Editor installed to exist able to make a TI-89 frequency distribution using these instructions.

Z Score TI 89: Steps

Instance trouble: Find the z score for α = .012 for a left-tailed examination on a standard normal distribution bend.

Step 1: Press Apps, scroll to the Stats/List Editor, and printing ENTER.
If you don't see the Stats/List Editor, you can download it here. Information technology's an official TI app and you'll need to transfer information technology to your calculator using the cable that originally came with your TI-89.

Step 2: Press F5 ii ane, to become to the Inverse Normal screen.

Step 3: Enter .012 in the Area box.

Step iv: Enter 0 for the mean, μ and 1 for the standard deviation, σ.

Step 5: Press ENTER.

Step 6: Read the consequence: the calculator should country "Changed = -two.25713". This is your z score.

Tip: If you lot are given a hateful and standard deviation, enter them in place of 0 and 1 in Step four.

That'southward how to detect a z score on the TI 89!

two. How to Observe a Z-Score in Excel

Z-Score in Excel: Overview

A z-score in Excel can apace be calculated using a basic formula. The formula for calculating a z-score is

z = (ten-μ) / σ,

where μ is the population mean and σ is the population standard deviation.

Note: if you don't know the population standard deviation or the sample size is below 6, you should use a t-score instead of a z-score.

Z-Score in Excel: Steps

Lookout man the video or read the steps below:

How to calculate a z score in Excel (all versions)

Example question: You take the GRE and scored 650 in the verbal section of the examination. Calculate a z-score in Excel to determine how good your score is compared to the general population of test takers. The population hateful (μ) is 469 and the population standard deviation (σ) is 119.

z-in-excel

Stride one: Blazon the population mean into a blank cell. For this example, type "469" into cell A2. Optional: type the word "mean" every bit a cavalcade header in prison cell A1 so you recollect what the value in cell A2 stands for.

Footstep 2: Enter the population standard deviation into a blank cell. For this example, type "119" into cell B2. Optional: blazon the words "standard departure" as a cavalcade header in cell B1 so you call back what the value in jail cell B2 stands for.

Step 3: Type the X-value (in this example problem, Ten is your GRE score) into a blank cell. For this example, type "650" into cell C2. Optional: blazon the words "X" as a column header in prison cell B1 so y'all remember what the value in cell B2 stands for.

Step four: Enter the following formula into an empty jail cell:
=(C2-A2)/B2

Step 5: Printing "Enter." The z-score will appear in cell D2: The z-score of 1.521008 in this case problem indicates your GRE score was 1.521008.

That's it! Y'all've found a z-score in Excel.

Tip: In one case you've entered the formula once, you tin use it over and over again. Merely type a new hateful, standard departure and Ten value into the relevant boxes.

References

Base of operations, C. (2018). Understanding Basic Statistics. Cengage Learning.
CDC. Anthropometric Reference Information for Children and Adults: United States, 2007–2010.
Salkind, North. (2016). Statistics for People Who (Retrieve They) Detest Statistics: Using Microsoft Excel 2022 Fourth Edition. SAGE Publications.

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