The z-score is a measure that indicates how far a raw score is from the mean in terms of standard deviation units. A positive z-score suggests that the value is above the mean, while a negative z-score indicates that it is below the mean. The z-score is commonly referred to as a standard score since it standardizes the distribution, enabling scores on various types of variables to be compared. A normally distributed dataset with a mean of 0 and a standard deviation of 1 is referred to as a standard normal distribution (SND).
Z-Score Formula:
In order to calculate a z-score, one must have information about the mean and standard deviation.
If the population mean and standard deviation are known, the formula for calculating the z-score is as follows:
z=\frac{x-\mu}{\sigma}
μ = population mean
σ = population standard deviation
x = raw score
When the population mean and standard deviation are unknown, the z score can be estimated using the sample mean and sample standard deviation. The modified z score formula is given as follows:
z=\frac{x-\bar{x}}{S}
x¯ = sample mean
S = sample standard deviation
x = raw score
Why Are Z-Scores Important?
It is useful to standardize 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 us to compare two scores that are from different samples (which may have different means and standard deviations).
How To Calculate
To calculate a z-score, the formula is z = (x-μ)/σ, where x represents the raw score, μ is the population mean, and σ is the population standard deviation.
The formula indicates that the z-score is obtained by subtracting the population mean from the raw score and dividing the result by the population standard deviation.
Spreadsheets
In order to calculate the z-score using a spreadsheet, you must first enter your data and compute the mean and standard deviation for the data set. This can be done using the following formulas:
= AVERAGE(A2:A7) = STDEV(A2:A7)
The data below has a mean of 24.6 and a standard deviation of 8.1.
A B C 1 Factor (x) Mean (μ) St. Dev.
A | B | C | |
| 1 | Factor (x) | Mean (μ) | St. Dev. (σ) |
| 2 | 10 | 24.6 | 8.1 |
| 3 | 32 | 24.6 | 8.1 |
| 4 | 18 | 24.6 | 8.1 |
| 5 | 27 | 24.6 | 8.1 |
| 6 | 21 | 24.6 | 8.1 |
| 7 | 15 | 24.6 | 8.1 |
By utilizing the z-score formula, you can calculate the z-score for each factor. To achieve this, apply the following formula to D2, D3, and so forth:
Cell D2: = (A2 – B2) / C2 Cell D3: = (A3 – B3) / C3
A | B | C | D | |
| 1 | Factor (x) | Mean (μ) | St. Dev. (σ) | Z-Score |
| 2 | 3 | 24.6 | 8.1 | -1.99 |
| 3 | 13 | 24.6 | 8.1 | -0.13 |
| 4 | 8 | 24.6 | 8.1 | -1.01 |
| 5 | 21 | 24.6 | 8.1 | 0.51 |
| 6 | 17 | 24.6 | 8.1 | -0.14 |
| 7 | 11 | 24.6 | 8.1 | -0.79 |
Z-Scores vs. Standard Deviation
| Represents the number of standard deviations a data point is from the mean. | Measures the amount of variability or dispersion within a set of data. |
| Calculated using the formula: z = (x – μ) / σ, where x is the data point, μ is the mean, and σ is the standard deviation. | Calculated using the formula: σ = √(Σ(x – μ)² / N), where x is each data point, μ is the mean, and N is the total number of data points. |
| A positive z-score indicates that the data point is above the mean, while a negative z-score indicates that the data point is below the mean. | A larger standard deviation indicates that the data is more spread out or dispersed, while a smaller standard deviation indicates that the data is more tightly clustered around the mean. |
| Z-scores are useful for comparing data from different sets or for comparing data within the same set. | Standard deviation is useful for identifying outliers and determining how closely the data resembles a normal distribution. |
Key Terms:
- The z-score value denotes the number of standard deviations away from the mean. A z-score of 0 corresponds to the mean.
- A positive z-score indicates that the raw score is greater than the mean. For instance, a z-score of +1 denotes that the value is 1 standard deviation above the mean.
- Conversely, a negative z-score indicates that the raw score is lower than the mean. A z-score of -2 signifies that the value is 2 standard deviations below the mean.
- Another approach to interpreting z-scores is by constructing a standard normal distribution, also known as the z-score distribution or probability distribution.
Standard Normal Distribution (SND)
- The standard normal distribution (SND), or z-distribution, shares the same shape as the raw score distribution. Therefore, if the raw scores follow a normal distribution, so do the z-scores. The mean of any SND is always zero.
- The standard deviation of any SND is always one.
- Thus, one standard deviation of the raw score (regardless of the actual value) corresponds to one z-score unit.
- The SND provides researchers with the ability to compute the likelihood of selecting a score randomly from the distribution (or sample). For example, there is a 68% chance of selecting a score between -1 and +1 standard deviations from the mean.
As depicted in Figure 4, the probability of randomly selecting a score ranging from -1.96 to +1.96 standard deviations from the mean is 95%. Therefore, if the possibility of randomly selecting a raw score is less than 5%, the outcome is considered statistically significant.
Calculating A Raw Score
To determine the corresponding raw score from a known z-score in a sample, we can use the following formula:
X = (z)(SD) + mean
In this formula, the z-score is multiplied by the standard deviation, and the resulting value is added to the mean to obtain the raw score.
It is reasonable to expect that a negative z-score corresponds to a raw score that is lower than the mean, while a positive z-score corresponds to a raw score that is higher than the mean.
For example, if the mean is 75 and the standard deviation is 5, and we have a z-score of -1.75, we can use the formula to find the corresponding raw score as:
X = (-1.75)(5) + 75 = 66.25
Therefore, the raw score that corresponds to a z-score of -1.75 is 66.25.
How to calculate z-score in excel?
To calculate the z-score of a specific value, x, you first need to calculate the mean and standard deviation of the sample. To find the mean, use the AVERAGE formula, and to find the standard deviation, use the STDEV.S formula.
For instance, if the range of scores in your sample begins at cell A1 and ends at cell A20, the formula =AVERAGE(A1:A20) returns the mean of those numbers, and the formula =STDEV.S(A1:A20) returns the standard deviation of those numbers.
Next, to calculate the z-score for a given value, use the formula: z = (x – mean) / standard deviation.
To simplify the formula, you can use cell references instead of directly writing the mean and standard deviation values. For example, = (A12 – B1) / C1.
To calculate the probability of 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), use the NORMSDIST function by typing =NORMSDIST(z-score) into a blank cell.
To calculate the probability of a larger z-score, which is the probability of observing a value greater than x (the area under the curve to the right of x), use the formula =1 – NORMSDIST(z-score).
Question 1
If the mean of a set is 22 and the standard deviation is 3, what is the z-score for a value of 28?
Putting the values in z-score formula, we get:
z = (28 – 22) / 3
z = 2
Therefore, the z-score for a value of 28 with a mean of 22 and a standard deviation of 3 is 2.
Question 2
If the mean of a set is 150 and the standard deviation is 8.3, what is the z-score for a value of 142.1?
Solution:
Putting the values into the z-score formula, we get:
z = (x – μ) / σ
z = (142.1 – 150) / 8.3
z = -0.901
Therefore, the z-score for a value of 142.1 in a set with a mean of 150 and a standard deviation of 8.3 is approximately -0.901.
Question 3
Let’s analyze how well did Sita perform in her English coursework compared to the other 50 students in the class. To answer this question, we can rephrase it as what percentage or number of students scored higher than Sita and what percentage or number of students scored lower than Sita? Let’s say Sita scored 85 out of 100, the mean score was 75, and the standard deviation was 10.
Score (x) | Mean (x̅) | Standard Deviation (σ) | |
| English Coursework | 85 | 75 | 10 |
We can calculate the z-score of Sita’s score using the formula:
z-score = (x – mean)/standard deviation
z-score = (85 – 75)/10
z-score = 1
Numerous real-world scenarios utilize z-scores, including medical assessments, test grading, business decision-making, and measuring investment and trading opportunities. Quantitative traders, also known as quant traders, use statistical measures like z-scores to evaluate potential trades.
When evaluating a normally distributed sample, a higher (or lower) z-score indicates that the data point is further away from the mean. The significance of this distance isn’t inherently positive or negative, but rather provides insight into the location of the data within the distribution.
Therefore, when evaluating an investment or opportunity, it ultimately comes down to personal preference. For instance, some investors might use a z-score range of -2.5 to 2.5 because 99% of normally distributed data falls within this range, whereas others may prefer scores closer to the mean and opt for a narrower range like -1.0 to 1.0.
Yes, a z-score can be negative. The z-score is a measure of how many standard deviations an individual data point is away from the mean. If a data point is below the mean, its z-score will be negative. Conversely, if a data point is above the mean, its z-score will be positive.
