Z scores (Z value) is the number of standard deviations a score or a value (x) is away from the mean. In other words, the Z-score measures the dispersion of data. Technically, a Z-score tells you how many standard deviations value (x) are below or above the population mean (µ). If the Z value is positive, it indicates that the value or score (x) is above the mean. Similarly, if the Z value is negative, it means the value (x) is below the mean.
What is a Standard Normal Distribution?
A Normal Standard Distribution curve is a symmetric distribution where the area under the normal curve is 1 or 100%. The standard normal distribution is a type of special normal distribution with a mean (µ) of 0 and a standard deviation of 1.
A standard normal distribution always has a mean of zero and has intervals that increase by 1. Each number on the horizontal line corresponds to the z-score. Hence, use Z Scores to transform a given standard distribution into something that is easy to calculate probabilities on as it can determine the likelihood of some event happening.
Any normal distribution with any value of mean (µ) and a sigma can be transformed into the standard normal distribution, where the mean of zero and a standard deviation of 1. This is also called standardization.
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A Z-score tells how much standard deviation a value or score is from the mean (µ). For example, if a Z-score is negative 3 means the value (x) is 3 standard deviations left of the mean. Similarly, if the Z-score is positive 2.5 means the value (x) is 2.5 standard deviations to the right of the mean (µ).
This is a common transformation, so there is a reference chart that allows us to look up values. Those values correlate to the value under the normal distribution curve – in other words, what’s the chance of an event happening? We use the Z table to find the percent chance.
How to Interpret the Z-score Table
Most importantly, the Z-score helps to calculate how much area that specific Z-score is associated with. A Z-score table is also known as a standard normal table used to find the exact area. The Z-score table tells the total quantity of area contained on the left side of any score or value (x).
Z Score Table Download
Using the Z score, find the percentage by using the formula: 1-NORMSDIST(Z), where Z is your calculated Z Score.
How to Calculate a Z Score by Hand
There are 2 different situations you need to be aware of when calculating a z score:
Z score for a sample
Z score for a population
While the z-score equations look very similar, remember that calculating the standard deviation of a population is different than the way you calculate the standard deviation of a sample.
The formula for transforming a score or observation x from any normal distribution to a standard normal score is :
Calculating a Z Score (generic)
Calculating a Z Score for a Population
Z Score for a Sample
How many parts in a population will be longer or greater than some number?
Z score examples using standard deviation
Example 1: Longer than / Greater than
Hospital stays for admitted patients at a certain hospital are measured in hours and were found to be normally distributed with an average of 200 hours and a standard deviation of 75 hours. How many of these stays can be expected to last for longer than 300 hours?
x=300
x̅ = 200
s=75
Z= x- x̅/s =(300-200)/75= 100/75= 1.33
Z score from the table for 1.33 = 0.9082
Since, we are looking for longer, solution is P(x>300) = P(Z>1.33) = 1- P(Z<1.33)= 1-0.9082 = 0.0918 = 9.18%
Z Score Positive Template
x=75
x̅ = 200
s=75
Z= x- x̅/s =(75-200)/75= -125/75= -1.667
Z score from the table for -1.667 = 0.0475
Since we are looking for less than, the solution is = 4.75%
Z Score Negative Template
P(X<175)
x=175
x̅ = 194
s=11.2
Z= x- x̅/s =(175-194)/11.2= -1.6964
Z_score from the table for -1.6964= 0.0455
P(X<175) = 4.55%
P(X<225)
x=225
x̅ =194
s=11.2
Z= x- x̅/s =(225-194)/11.2= 2.7678
Z score from the table for 2.7678 = 0.9971=99.71%
Since we are looking for weights between 175 and 225, P(175<x<225) = 99.71%-4.55% = 95.16%
Question: This formula Z = (X – μ)/σ is used to calculate a Z score that, with the appropriate table, can tell a Belt what ____________________________________.
A) Ratio the area under the curve to the total population
B) Number of Standard Deviations between X and μ
C) The Median of the sample population is
D) Proportion of the data is between X and μ
Answer:
B: 1.33 This is an easy algebra question. Z = (X – μ)/σ = (28-32) / 3 = 4/3 = 1.33. See Z Scores.
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