Want to find a z-score from a probability? Use the inverse normal distribution calculator.
What Is a Raw Score?
A raw score is the original value before it is converted into a z-score, percentile, standard score, or another transformed score. For example, if a student scores 82 on an exam, then 82 is the raw score. However, if 82 is later converted into a z-score or percentile rank, those are standardized forms of the original score.
Raw scores are important in statistics because they help us understand values in their original measurement units. For instance, a z-score of -1.5 for students’ weights means the value is 1.5 standard deviations below the mean. However, the z-score does not directly tell us the actual weight. When we calculate the raw score, we convert the standardized value back into the original unit of measurement, such as kilograms or pounds.
Therefore, a raw score calculator helps you convert a z-score, percentile, or probability back into the original x value, especially when you already know the mean and standard deviation.
How to Use the Raw Score Calculator
This raw score calculator helps you find a raw score, also called x, from a z-score, a percentile, or a right-tail probability. Besides giving you the final answer, it also shows the step-by-step solution so you can see exactly how the raw score was obtained.
To find a raw score using the calculator:
- Select whether you want to find the raw score from a z-score, left-tail probability, or right-tail probability.
- Enter the mean and standard deviation.
- If you selected the z-score option, enter the known z-score.
- If you selected left-tail or right-tail probability, enter the known probability or percentile either as a percent or a decimal.
- Click Calculate.
The calculator will return the raw score and show you how the raw score (x) was computed using the formula.
Example 1: Find a Raw Score from a Z-Score
Suppose exam scores are normally distributed with a mean of 70 and a standard deviation of 8. What raw score corresponds to a z-score of 1.25?
To find this raw score using the calculator:
- Select Z-score.
- Enter the mean, μ = 70.
- Enter the standard deviation, σ = 8.
- Enter the z-score, z = 1.25.
- Click Calculate.
The calculator will instantly return: Raw score, x = 80
Example 2: Find a Raw Score from a Percentile
Suppose test scores are normally distributed with a mean of 70 and a standard deviation of 8. What raw score is at the 90th percentile?
The 90th percentile means 90% of the scores fall below that value. This is the same as a left-tail probability of 0.90.
To find this raw score using the calculator:
- Select the Left-tail probability option.
- Enter the mean, μ = 70.
- Enter the standard deviation, σ = 8.
- Enter the probability as 90 if using percent format, or 0.90 if using decimal format.
- Click Calculate.
The calculator will find the z-score for the 90th percentile and then convert it into a raw score. Thus, the raw score will be: x = 80.25. This means that about 90% of scores are below 80.25, assuming the scores follow a normal distribution.
Example 3: Find a Raw Score from a Right-Tail Probability
Suppose scores are normally distributed with a mean of 100 and a standard deviation of 15. What raw score is exceeded by only 5% of the distribution?
This is a right-tail probability question because we are looking for the value below which 5% of scores fall.
To find this raw score using the calculator:
- Select the Right-tail probability option.
- Enter the mean, μ = 100.
- Enter the standard deviation, σ = 15.
- Enter the probability as 5 if using percent format, or 0.05 if using decimal format.
- Click Calculate.
The calculator will convert the right-tail probability into the matching left-tail probability, find the corresponding z-score, and then calculate the raw score. The results will be x = 124.67. This means about 95% of values are below 124.67, while about 5% are above it.
Raw Score Formula
By definition, the raw score formula is: Raw score = Mean + (Z-score × Standard deviation)
In symbols, the formula is: x = μ + zσ
Where:
- x is the raw score
- μ is the mean
- z is the z-score
- σ is the standard deviation
The raw score formula is derived from the z-score formula by making x the subject of the formula.
Note. If you want to find a raw score from a percentile or probability, you’ll need to convert the probability into a z-score first before applying the raw score formula.
Example 1: Finding a Raw Score (x) from a Z-Score
Suppose a distribution has a mean of 70 and a standard deviation of 8. Find the raw score that corresponds to a z-score of 1.25.
Solution
By definition, the formula for finding a raw score (x) from z is: x = μ + zσ
From the question:
- Mean, μ = 70
- Standard Deviation, σ = 8
- Z-score = 1.25
Substituting the values into the formula gives: x = 70 + 1.25 (8)
=80
Therefore, the raw score is x = 80.
Example 2: Finding a Raw Score (x) from a Percentile
Suppose test scores have a mean of 70 and a standard deviation of 8. Find the raw score at the 90th percentile.
Solution
In the standard normal distribution, the 90th percentile means the left-tail probability is 0.90.
Therefore, we need to find a raw score value, x, such that P(X<x) = 0.90, given that the mean, μ = 70, and standard deviation, σ = 8.
To find the value of x, we need to first find the z-score value corresponding to a left-tail probability of 0.90. Using the standard normal distribution table, the z-score value corresponding to a left-tail probability of 0.90 is 1.28.
Since we have the z-score, mean, and standard deviation, we can now apply the raw score formula to find the value of x.
Thus, x = μ + zσ
= 70 + 1.28 (8)
= 80.24
Therefore, the raw score at the 90th percentile is approximately 80.24.
Example 3: Finding a Raw Score (x) from a Right-Tail Probability
Suppose scores have a mean of 100 and a standard deviation of 15. Find the raw score that is exceeded by only 5% of the distribution.
Solution
Here, we need to find the value of x such that P(X>x) = 0.05.
To find the value of x, we first need to convert this right-tail probability to a left-tail probability because the z-score table is left-sided. Thus, P(X>x) = 0.05 is equivalent to P(X<x) = 1-0.05.
Thus, we need to find the value of x such that P(X<x) = 0.95. Looking up the value of 0.95 in the table gives 1.645.
Now that we have the z-score, mean, and standard deviation, we can comfortably compute the value of x as follows:
By definition, x = μ + zσ
= 100 + 1.645 (15)
=124.675
Therefore, the raw score exceeded by only 5% of the distribution is approximately 124.675.
When Should You Use a Raw Score Calculator?
Use a raw score calculator when you want to convert a standardized value back into the original scale.
This is useful when you need to:
- Convert a z-score into a raw score
- Find an exam cutoff score
- Find a score from a percentile
- Find a score from a normal probability
- Convert standardized results into original units
- Interpret scores in a more meaningful way
Frequently Asked Questions
A raw score is the original value before it is converted into a z-score, percentile, or standardized score. For example, if a student scores 76 on a test, 76 is the raw score.
To find a raw score (x) from a z-score, we use the formula: x = μ + zσ, where μ is the mean, z is the z-score value, and σ is the standard deviation.
Yes. In this case, you need to first convert a percentile into a z-score, and then the z-score into a raw score. Want to convert a percentile to a z-score? Use the percentile to z-score calculator.
No. A raw score is the actual value, while a percentile tells you the percentage of values below that score. For example, a raw score of 80 may be the 90th percentile if 90% of values are below 80.
The mean and standard deviation define the normal distribution. Without them, the calculator cannot convert a z-score or probability into a raw score on the original scale.
A left-tail probability gives the area below a value, while a right-tail probability gives the area above a value. For example, the 90th percentile is a left-tail probability of 0.90, while a right-tail probability of 0.10 means 10% of values are above the score.
