___________ Score Lower on Measures of Family Solidarity and Identity Than ___________.
Z-Score: Definition, Calculation and Interpretation
Past Dr. Saul McLeod, published 2019
What does the z-score tell y'all?
A z-score describes the position of a raw score in terms of its distance from the hateful, when measured in standard deviation units. The z-score is positive if the value lies above the mean, and negative if it lies below the mean.
It is also known as a standard score, considering information technology allows comparison of scores on different kinds of variables by standardizing the distribution. A standard normal distribution (SND) is a commonly shaped distribution with a mean of 0 and a standard deviation (SD) of 1 (run across Fig. ane).
Figure 1. A standard normal distribution (SND).
Why are z-scores important?
It is useful to standardized the values (raw scores) of a normal distribution by converting them into z-scores because:
(a) it allows researchers to calculate the probability of a score occurring within a standard normal distribution;(b) and enables u.s.a. to compare two scores that are from unlike samples (which may have dissimilar means and standard deviations).
How do yous calculate the z-score?
The formula for calculating a z-score is is z = (x-μ)/σ, where x is the raw score, μ is the population hateful, and σ is the population standard departure.
As the formula shows, the z-score is but the raw score minus the population mean, divided by the population standard divergence.
Figure ii. Z-score formula in a population.
When the population mean and the population standard departure are unknown, the standard score may be calculated using the sample mean (x̄) and sample standard deviation (s) every bit estimates of the population values.
How exercise yous translate a z-score?
The value of the z-score tells you lot how many standard deviations you are away from the mean. If a z-score is equal to 0, it is on the mean.
A positive z-score indicates the raw score is higher than the hateful average. For example, if a z-score is equal to +one, it is i standard deviation higher up the mean.
A negative z-score reveals the raw score is beneath the mean average. For instance, if a z-score is equal to -2, it is 2 standard deviations below the mean.
Another mode to interpret z-scores is by creating a standard normal distribution (also known as the z-score distribution or probability distribution).
Fig 3 illustrates the of import features of any standard normal distribution (SND).
Effigy 3. A standard normal distribution (SND) and normal distribution.
- The SND (i.eastward. z-distribution) is always the same shape every bit the raw score distribution. For example, if the distribution of raw scores if usually distributed, and then is the distribution of z-scores.
- The hateful of any SND always = 0.
- The standard deviation of any SND e'er = 1. Therefore, ane standard departure of the raw score (whatsoever raw value this is) converts into 1 z-score unit of measurement.
The SND allows researchers to calculate the probability of randomly obtaining a score from the distribution (i.due east. sample). For instance, at that place is a 68% probability of randomly selecting a score between -ane and +i standard deviations from the mean (see Fig. four).
Figure 4. Proportion of a standard normal distribution (SND) in percentages.
The probability of randomly selecting a score between -1.96 and +1.96 standard deviations from the mean is 95% (see Fig. four). If there is less than a 5% chance of a raw score being selected randomly, then this is a statistically pregnant consequence.
Learn how to use a z-score table
How to summate a raw score when a z-score is known
Sometimes we know a z-score and want to find the corresponding raw score. The formula for calculating a z-score in a sample into a raw score is given below:
X = (z)(SD) + hateful
As the formula shows, the z-score and standard deviation are multiplied together, and this figure is added to the mean.
Check your answer makes sense: If nosotros have a negative z-score the corresponding raw score should be less than the mean, and a positive z-score must stand for to a raw score college than the mean.
How to calculate a z-score using excel
To summate the z-score of a specific value, x, showtime you must calculate the mean of the sample by using the AVERAGE formula.
For example, if the range of scores in your sample brainstorm at jail cell A1 and end at cell A20, the formula =Average(A1:A20) returns the average of those numbers.
Side by side, you mush calculate the standard deviation of the sample by using the STDEV.South formula. For example, if the range of scores in your sample brainstorm at cell A1 and end at cell A20, the formula = STDEV.S (A1:A20) returns the standard deviation of those numbers.
Now to calculate the z-score type the following formula in an empty prison cell: = (x – mean) / [standard deviation].
To brand things easier, instead of writing the hateful and SD values in the formula you could utilize the cell values respective to these values. For example, = (A12 – B1) / [C1].
Then, to calculate the probability for a SMALLER z-score, which is the probability of observing a value less than x (the area under the curve to the LEFT of x), blazon the following into a bare cell: = NORMSDIST( and input the z-score you lot calculated).
To find the probability of LARGER z-score, which is the probability of observing a value greater than ten (the area under the bend to the RIGHT of x), blazon: =one - NORMSDIST (and input the z-score you lot calculated).
How to reference this commodity:
How to reference this article:
McLeod, S. A. (2019, May 17). Z-score: definition, calculation and interpretation. Just Psychology. world wide web.simplypsychology.org/z-score.html
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