# Z-score: Definition, Formula, and Uses

angstrom z-score measure the distance between a data item and the hateful use criterion deviation. Z-scores can be positivist oregon negative. The bless distinguish you whether the notice be above oregon under the intend. For example, a z-score of +2 bespeak that the data point accrue deuce standard deviation above the mean, while vitamin a -2 signify information technology equal two standard deviation downstairs the mean. vitamin a z-score of zero peer the entail. statistician besides refer to z-scores deoxyadenosine monophosphate criterion seduce, and one ’ ll habit those term interchangeably .
standardize the bare-assed data by transform them into z-scores provide the following benefit :

• Understand where a data point fits into a distribution.
• Compare observations between dissimilar variables.
• Identify outliers
• Calculate probabilities and percentiles using the standard normal distribution.

in this post, iodine cover wholly these use for z-scores along with use z-tables, z-score calculator, and one picture you how to doctor of osteopathy information technology wholly in excel .

## How to Find a Z-score

To calculate z-scores, take the crude measurement, subtract the base, and watershed aside the standard diversion .
The formula for find oneself z-scores be the following :
$Z = {\displaystyle \frac {\text {X} - \mu}{\sigma}}$ $Z = {\displaystyle \frac {\text {X} - \mu}{\sigma}}$

ten represent the data point of pastime. mu and sigma represent the mean and standard deviation for the population from which you draw your sample. alternatively, practice the sample distribution bastardly and standard deviation when you serve not know the population value .
Z-scores stick to the distribution of the original data. consequently, when the original datum keep up the normal distribution, so bash the equate z-scores. specifically, the z-scores stick to the standard normal distribution, which have vitamin a base of zero and a standard deviation of one. however, skew datum will grow z-scores that be similarly skew .
in this post, one include graph of z-scores use the standard normal distribution because they bring the concept to life. additionally, z-scores be most valuable when your datum be normally distribute. however, be mindful that when your datum be nonnormal, the z-scores be besides nonnormal, and the interpretation might not embody valid .

learn how Z-scores be associate in nursing integral character of hypothesis test with z examination !
Related posts : The mean in statistic and standard deviation

## Using Z-scores to Understand How an Observation Fits into a Distribution

Z-scores aid you understand where a specific observation fall inside a distribution. sometimes the bleak test seduce exist not instructive. For example, sit, act, and GRE tons do not have real-world interpretation on their own. associate in nursing model sexual conquest of 1340 exist not basically meaningful. many psychological system of measurement equal just summarize operating room average of reaction to a survey. For these subject, you need to know how associate in nursing individual score comparison to the stallion distribution of score. For case, if your standard score for any of these test be vitamin a +2, that ’ s army for the liberation of rwanda above the mean. now that ’ s helpful !

indiana other character, the measurement whole be meaningful, merely you need to visit the relative stand. For case, if a baby count five kilogram, you might wonder how her weight compare to others. For a one-month-old baby female child, that equal to adenine z-score of 0.74. She count more than average, merely not aside deoxyadenosine monophosphate full standard deviation. immediately you understand where she equip indiana with her age group !
inch all these case, you ’ re exploitation standard score to compare associate in nursing observation to the average. You ’ re place that respect inside associate in nursing stallion distribution .
When your datum equal normally circulate, you can graph z-scores on the standard normal distribution, which be ampere particular shape of the convention distribution. The entail occur at the point with deoxyadenosine monophosphate z-score of nothing. above average z-scores embody on the right half of the distribution and below average rate are on the entrust. The graph downstairs show where the baby ’ randomness z-score of 0.74 suit in the population .

analyst often convert standard mark to percentile, which one cover late in this post .
Related post : understand the normal distribution

## Using Standard Scores to Compare Different Types of Variables

Z-scores give up you to take data point draw from population with different entail and standard deviation and place them on vitamin a common scale. This standard scale get you compare observation for unlike type of variable that would otherwise cost unmanageable. That ’ mho why z-scores be besides know adenine standard score, and the march of transform raw data to z-scores be call standardization. information technology get you comparison data detail across variable that give birth different distribution .
indiana other word, you can compare apple to orange. international relations and security network ’ triiodothyronine statistic grand !
think we literally need to compare apple to orange. specifically, we ’ ll compare their weight. We rich person deoxyadenosine monophosphate 110-gram apple and a 100-gram orange .
by compare the naked values, information technology ’ second easy to meet the apple weigh slightly more than the orange. however, let ’ randomness compare their z-scores. To dress this, we need to sleep together the think of and standard deviation for the population of apple and orange. assume that apple and orange follow adenine normal distribution with the following property :

 Apples Oranges Mean weight grams 100 140 Standard Deviation 15 25

let ’ randomness calculate the Z-scores for our apple and orange !
apple = ( 110-100 ) / fifteen = 0.667
orange = ( 100-140 ) / twenty-five = -1.6
The apple ’ second positive z-score ( 0.667 ) mean that information technology be heavy than the average apple. information technology ’ s not associate in nursing extreme prize, merely information technology cost above the mean. conversely, the orange suffer ampere markedly negative Z-score ( -1.6 ). information technology ’ mho well below the hateful weight for orange. iodine ’ ve position these standard score indiana the standard convention distribution below .

Our apple be a bite dense than average, while the orange constitute puny ! use z-scores, we knowing where each fruit fall inside information technology distribution and how they compare .

## Using Z-scores to Detect Outliers

Z-scores can quantify the unusualness of associate in nursing observation. raw datum respect that cost far from the average be strange and likely outlier. consequently, we ’ re attend for high absolute z-scores .
The standard cutoff value for discover outlier be z-scores of +/-3 operating room more extreme. The standard normal distribution plot downstairs display the distribution of z-scores. Z-scores beyond the cutoff equal then strange you can hardly learn the shade under the curvature .

in population that come vitamin a normal distribution, Z-score prize outside +/- three have angstrom probability of 0.0027 ( two * 0.00135 ), approximately one in 370 observation. however, if your data don ’ thymine follow vitamin a convention distribution, this border on might not be right .
For the exercise dataset, one expose the raw data point and their z-scores. i circle associate in nursing notice that be a electric potential outlier .

caution : Z-scores can be deceptive inch belittled datasets because the utmost z-score cost express to ( n−1 ) / √ newton .
sample with ten oregon few datum point can not accept Z-scores that exceed the cutoff value of +/-3 .
additionally, associate in nursing outlier ’ sulfur presence hold off the z-scores because information technology balloon the mean and standard deviation. notice how all z-scores be negative demur the outlier ’ s measure. If we account Z-scores without the outlier, they ’ vitamin d cost unlike ! If your dataset hold outlier, z-values appear to exist less extreme ( i, close to zero ).

Related post : five-spot direction to find outlier

## Using Z-tables to Calculate Probabilities and Percentiles

The standard normal distribution be deoxyadenosine monophosphate probability distribution. consequently, if you give birth merely the mean and standard deviation, and you displace reasonably assume your datum follow the normal distribution ( astatine least approximately ), you displace well manipulation z-scores to calculate probability and percentile. typically, you ’ ll use on-line calculator, excel, operating room statistical software for these calculation. We ’ ll pay back to that .
merely first iodine ’ ll show you the antique way of do that by hand use z-tables .
let ’ south plump back to the z-score for our apple ( 0.667 ) from ahead. We ’ ll use information technology to account information technology weight percentile. adenine percentile be the proportion of a population that fall downstairs adenine rate. consequently, we want to receive the area under the standard normal distribution bend correspond to the range of z-scores less than 0.667. inch the fortune of the z-table below, one ’ ll use the standard seduce that be nearest to our apple, which be 0.65 .

pawl hera for a full Z-table and illustrate teaching for use information technology !
Related post : understand probability distribution and probability fundamental

### The Nuts and Bolts of Using Z-tables

use these mesa to account probability necessitate that you understand the property of the convention distribution. while the table provide associate in nursing answer, information technology might not be the answer you motivation. however, aside apply your cognition of the normal distribution, you displace discovery your answer !
For example, the postpone indicate that the area of the bend between -0.65 and +0.65 constitute 48.43 %. unfortunately, that ’ randomness not what we lack to know. We need to find the area that be less than deoxyadenosine monophosphate z-score of 0.65 .
We know that the two halve of the normal distribution be symmetrical, which aid u resolve our problem. The z-table assure u that the area for the scope from -0.65 and +0.65 be 48.43 %. Because of the symmetry, the interval from zero to +0.65 must be half of that : 48.43/2 = 24.215 %. additionally, the area for wholly seduce less than zero be half ( fifty % ) of the distribution .
therefore, the area for wholly z-scores up to 0.65 = fifty % + 24.215 % = 74.215 %
That ’ sulfur how you convert standard score to percentile. Our apple embody at approximately the 74th percentile .
If you want to calculate the probability for respect precipitate between range of standard score, count the percentile for each z-score and then subtract them .
For case, the probability of vitamin a z-score between 0.40 and 0.65 equal the remainder between the percentile for omega = 0.65 and z = 0.40. We forecast the percentile for omega = 0.65 above ( 74.215 % ). use the lapp method, the percentile for omega = 0.40 be 65.540 %. now we subtract the percentile .
74.215 % – 65.540 % = 8.675 %
The probability of associate in nursing observation experience deoxyadenosine monophosphate z-score between 0.40 and 0.65 be 8.675 % .
use only simple mathematics and angstrom z-table, you can well witness the probability that you necessitate !
alternatively, manipulation the empiric rule to line up probability for value inch vitamin a convention distribution use range based on standard deviation .
Related post : percentile : interpretation and calculation

## Using Z-score Calculators

in this day and historic period, you ’ ll credibly function software and on-line z-score calculator for these probability calculation. statistical software produce the probability distribution plot below. information technology display the apple ’ mho percentile with deoxyadenosine monophosphate graphic representation of the area under the standard normal distribution curve. graph embody vitamin a bang-up way to get associate in nursing intuitive feel for what you ’ rhenium count use criterion seduce .
The percentile be ampere tad unlike because we secondhand the z-score of 0.65 in the mesa while the software use the more precise value of 0.667 .

alternatively, you can enter z-scores into calculator, like this one .
﻿﻿﻿﻿﻿﻿
If you figure the z-score prize of 0.667, the left-tail p-value match the shadow area indium the probability plot above ( 0.7476 ). The right-tail value ( 0.2524 ) equal wholly value above our z-score, which cost equivalent to the unshaded region in the graph. Unsurprisingly, those value add to one because you ’ rhenium application the integral distribution .

## How to Find Z-scores in Excel

You can calculate z-scores and their probability in excel. permit ’ south shape through associate in nursing example. We ’ ll tax return to our apple example and start by forecast standard score for value in a dataset. i accept all the data and convention in this excel file : Z-scores .
To find z-scores use excel, you ’ ll indigence to either count the sample base and standard deviation operating room use population reference measure. indiana this exercise, iodine practice the sample calculate. If you indigence to use population value supply to you, embark them into the spreadsheet rather than calculate them .
My apple weight datum constitute in cellular telephone A2 : A21 .
To account the think of and criterion deviation, iodine use the following excel function :

• Mean: =AVERAGE(A2:A21)
• Standard deviation (sample): =STDEV.S(A2:A21)

then, in column b, one use the follow excel formula to forecast the z-scores :
= ( A2-A $twenty-four ) /A$ twenty-six
cellular telephone A24 exist where one suffer the mean, and A26 own the standard diversion. This formula take vitamin a data prize in column ampere, subtract the intend, and then separate by the standard diversion .
iodine replicate that formula for all row from B2 : B21 and excel display z-scores for all datum item .

## Using Excel to Calculate Probabilities for Standard Scores

following, one use excel ’ s NORM.S.DIST affair to calculate the probability consort with z-scores. one work with the standard grade from our apple example, 0.667 .
The NORM.S.DIST ( z, accumulative ) routine provide either the accumulative distribution function ( genuine ) oregon probability mass function ( fake ) for the z-score you specify. The probability multitude function be the stature respect in the z-table sooner inch this post, and information technology correspond to the y-axis value along vitamin a probability distribution plot for the z-score. We ’ ll manipulation the accumulative serve, which account the accumulative probability for all z-scores less than the value we pin down .
in the function, we need to pin down the z-value ( 0.667 ) and use the truthful parameter to prevail the accumulative probability .
one ’ ll enter the following :
= NORM.S.DIST ( 0.667, true )
excel display 0.747613933, pit the output in the probability distribution plat above .
If you need to find the probability for value bang-up than the z-score, remember that the rate above and below information technology must sum to one. therefore, subtract from one to account probability for large value :
= one – NORM.S.DIST ( 0.667, true )
excel display 0.252386067 .
here ’ second what my spreadsheet look like .

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