Standard Scores and Deviation Scores | An Easy-to-Understand Psychology Statistics Example
In this article, Standard Scores and Deviation Values This article explains the topic using a basic example from psychological statistics.
In statistics, it is important not merely to look at “what score was obtained” but to consider how far a score is from the mean and where a person stands within the group where that score is located within the group. Representative indicators used for this purpose include standard score and deviation score .
We will consider the following example.
Example Problem
A test was taken by 200 people The scores had a mean of 58 points and a standard deviation of 8 .
(1) Calculate the standard score and deviation score of a person who scored 50 on this test.
(2) Assuming the data are normally distributed, calculate how many people scored 50 or higher.
(3) Calculate the score corresponding to the cutoff for the top 25% on this test.
What Is a Standard Score?
A standard score indicates how far a particular score is from the mean the number of standard deviations from the mean This value represents the position of a score relative to the mean. It is usually z-score also called
The formula for the standard score is as follows.
z = (X - M) / SD
Here, X is the individual score, M is the mean, and SD is the standard deviation.
Using a standard score makes it possible to compare distances from the mean on a common scale even when the original scores use different units.
What Is a Deviation Score?
A deviation score is a transformed standard score designed so that mean 50, standard deviation 10 a value transformed to have this standard scale.
The formula is as follows.
Deviation score = 50 + 10 × z
Because a standard score has a mean of 0, it can be difficult to understand intuitively at first. A deviation score rescales it so that the mean is 50 and the result is easier to interpret.
(1) Standard Score and Deviation Score for a Person Who Scored 50
First, calculate the standard score of the person who scored 50.
The mean is 58 and the standard deviation is 8, so
z = (50 - 58) / 8 = -8 / 8 = -1
Therefore, the standard score for a score of 50 is -1 .
Next, calculate the deviation score.
Deviation score = 50 + 10 × (-1) = 40
Therefore, the deviation score of a person who scored 50 points is 40 becomes
(2) How many people scored 50 points or higher?
Next, assuming that scores follow a normal distribution, calculate the number of people who scored 50 or higher.
We already know that the standard score for 50 points is z = -1 .
In the standard normal distribution, the probability above z = -1 we only need to consider this probability.
From the standard normal distribution table, the area to the left of z = -1 is approximately 0.1587 Therefore, the probability of z ≥ -1 is
1 - 0.1587 = 0.8413
becomes
Because there are 200 examinees, the number scoring 50 or higher is
200 × 0.8413 = 168.26
which is approximately 168 people can be estimated as
How to Read the Standard Normal Distribution Table
In this problem, the standard normal distribution table this probability is found using
A standard normal distribution table is used to find the cumulative probability up to a given standard score z. Because it is used frequently in statistics, it is important to become familiar with reading the table along with calculating standard scores.
A person who scored 50 is one standard deviation below the mean, so approximately 84.13% of the distribution lies above that person.
(3) Score Corresponding to the Cutoff for the Top 25%
Finally, calculate the score corresponding to the cutoff for the top 25%.
Being in the top 25% means the 75th percentile from the bottom This means finding the score located at
Looking up the z value corresponding to a cumulative probability of 0.75 in the standard normal table gives approximately 0.67 becomes
To convert this back to the original score,
X = M + z × SD
use
Here, the mean is 58, the standard deviation is 8, and z = 0.67, so
X = 58 + 0.67 × 8 = 58 + 5.36 = 63.36
Therefore, the score corresponding to the top-25% cutoff is approximately 63.4 points .
In other words, people scoring about 63.4 points or higher can be considered to be in the top 25%.
Key Points to Learn from This Example
This example covers not only how to calculate standard scores and deviation scores but also Using the Normal Distribution and Standard Normal Distribution Table is also an important point.
First, calculating the standard score shows how far an individual score is from the mean in standard-deviation units. Converting that standard score into a deviation score then produces a more intuitive value.
Furthermore, assuming a normal distribution makes it possible to solve problems such as the number of people scoring above a certain value or the score corresponding to a particular upper percentile. This approach is used frequently not only in psychological statistics but also in educational measurement and analysis of test results.
Summary
In this example, standard scores and deviation scores were calculated under the conditions of a mean of 58 points, a standard deviation of 8, and 200 examinees.
The standard score of a person who scored 50 points is -1 and the deviation score is 40 The result is as shown. Assuming a normal distribution, the number scoring 50 or higher is approximately 168 people and the cutoff score for the top 25% is approximately 63.4 points respectively.
Standard scores and deviation scores are important indicators for locating an individual’s score within a group. When studying statistics and psychology, it is important to understand not only the formulas but also how to interpret a deviation from the mean Understanding the underlying idea itself is important.
