Finding the Z score is a crucial skill in statistics, enabling you to understand where a particular data point stands in relation to the rest of its distribution. HOW.EDU.VN provides expert guidance on this topic, offering precise methods and tools to calculate and interpret Z scores accurately. By understanding the Z score, you can compare data from different distributions and make informed decisions; leverage our expertise to master the Z score calculation and its applications. Explore concepts like standard deviation and normal distribution to enhance your understanding.
1. Understanding the Z Score: Definition and Significance
The Z score, also known as the standard score, is a statistical measure that quantifies the distance between a data point and the mean of a dataset in terms of standard deviations. It serves as a critical tool for standardizing data, allowing for meaningful comparisons across different datasets and distributions. The Z score transforms raw data into a standard scale, making it easier to interpret and compare values regardless of their original units.
1.1. Defining the Z Score
The Z score is a dimensionless quantity that indicates how many standard deviations a data point is above or below the mean. A positive Z score indicates that the data point is above the mean, while a negative Z score indicates it is below the mean. A Z score of zero means the data point is exactly at the mean.
The formula for calculating the Z score is:
z = (x – μ) / σ
Where:
- z is the Z score
- x is the raw data point
- μ is the population mean
- σ is the population standard deviation
For a sample, the formula is similar:
z = (x – x̄) / s
Where:
- x̄ is the sample mean
- s is the sample standard deviation
1.2. Importance of Z Score in Statistics
The Z score is essential in statistics for several reasons:
- Standardization of Data: The Z score standardizes data, allowing for comparisons between different datasets with different units and scales. This is particularly useful when dealing with data from various sources or when combining multiple datasets.
- Identifying Outliers: Z scores help identify outliers in a dataset. Data points with very high or very low Z scores (typically, |Z| > 3) are considered outliers, as they deviate significantly from the mean.
- Calculating Probabilities: Z scores are used to calculate probabilities associated with specific data points. By referencing a Z-table (also known as a standard normal table), you can determine the probability of a data point falling within a certain range of the distribution.
- Hypothesis Testing: Z scores are used in hypothesis testing to determine whether a sample mean is significantly different from a population mean. This is a fundamental aspect of statistical inference.
- Quality Control: In manufacturing and other industries, Z scores are used to monitor process control. Deviations from the expected Z score can indicate problems in the production process, prompting corrective actions.
- Comparative Analysis: Z scores facilitate comparative analysis by providing a standardized metric for evaluating performance or outcomes across different groups or individuals. For example, comparing test scores from different schools or evaluating the effectiveness of different treatments in a clinical trial.
1.3. Real-World Applications of Z Scores
Z scores have numerous applications across various fields:
- Education: Standardizing test scores to compare student performance across different tests.
- Healthcare: Assessing patient health metrics relative to population norms.
- Finance: Evaluating investment performance compared to market benchmarks.
- Manufacturing: Monitoring product quality and identifying deviations from standards.
- Sports: Analyzing athlete performance and comparing individuals across different events.
- Environmental Science: Assessing pollution levels compared to established thresholds.
By understanding and applying Z scores, professionals in these fields can make informed decisions and gain valuable insights from their data.
2. Prerequisites for Calculating Z Score
Before diving into the calculation of Z scores, it’s essential to grasp the underlying concepts and ensure you have the necessary data. This involves understanding normal distributions, mean, and standard deviation, as well as gathering the correct data points.
2.1. Understanding Normal Distribution
The normal distribution, also known as the Gaussian distribution, is a fundamental concept in statistics. It is a symmetrical, bell-shaped distribution characterized by its mean (μ) and standard deviation (σ). Many natural phenomena tend to follow a normal distribution, making it a valuable tool for statistical analysis.
Key properties of the normal distribution include:
- Symmetry: The distribution is symmetrical around the mean, meaning the left and right sides are mirror images of each other.
- Bell-Shaped Curve: The distribution has a distinct bell shape, with the highest point at the mean and gradually decreasing as you move away from the mean.
- Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal.
- Empirical Rule: The empirical rule (or 68-95-99.7 rule) states that approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
Understanding the normal distribution is crucial because the Z score is based on the assumption that the data follows this distribution. When data is normally distributed, Z scores can be used to accurately determine the probability of a data point occurring within a specific range.
2.2. Defining Mean and Standard Deviation
The mean and standard deviation are two essential statistical measures that describe the central tendency and variability of a dataset.
-
Mean (μ): The mean is the average of all data points in a dataset. It is calculated by summing all the values and dividing by the number of values. The mean represents the center of the distribution.
μ = (Σx) / N
Where:
- Σx is the sum of all data points
- N is the number of data points
-
Standard Deviation (σ): The standard deviation measures the spread or dispersion of data points around the mean. A low standard deviation indicates that the data points are clustered closely around the mean, while a high standard deviation indicates that the data points are more spread out.
σ = √[(Σ(x – μ)^2) / N]
Where:
- x is each individual data point
- μ is the mean of the dataset
- N is the number of data points
For a sample, the formulas are slightly different:
-
Sample Mean (x̄): The sample mean is the average of all data points in a sample.
x̄ = (Σx) / n
Where:
- Σx is the sum of all data points in the sample
- n is the number of data points in the sample
-
Sample Standard Deviation (s): The sample standard deviation measures the spread of data points around the sample mean.
s = √[(Σ(x – x̄)^2) / (n – 1)]
Where:
- x is each individual data point in the sample
- x̄ is the sample mean
- n is the number of data points in the sample
2.3. Gathering Required Data
To calculate the Z score, you need the following data:
- Raw Data Point (x): The specific value you want to standardize or compare.
- Population Mean (μ) or Sample Mean (x̄): The average of the entire population or sample.
- Population Standard Deviation (σ) or Sample Standard Deviation (s): A measure of the spread of the data around the mean.
Ensure that you use the appropriate formulas for the population or sample, depending on whether you have data for the entire population or just a sample. Accurate data is critical for calculating meaningful Z scores.
3. Step-by-Step Guide to Calculating Z Score
Calculating the Z score involves a straightforward process once you have the necessary data. This section provides a detailed, step-by-step guide to ensure accurate calculations.
3.1. Step 1: Identify the Raw Score (x)
The first step in calculating the Z score is to identify the raw score (x) you want to standardize. This is the specific data point you are interested in comparing to the rest of the dataset.
For example, if you want to find the Z score for a student who scored 85 on a test, the raw score (x) is 85.
3.2. Step 2: Determine the Mean (μ or x̄)
Next, determine the mean of the dataset. If you have data for the entire population, use the population mean (μ). If you are working with a sample, use the sample mean (x̄).
Calculating the mean involves summing all the data points and dividing by the number of data points.
-
Population Mean (μ):
μ = (Σx) / N
-
Sample Mean (x̄):
x̄ = (Σx) / n
For example, if the test scores for all students (population) have a mean of 75, then μ = 75. If you only have the scores for a subset of students (sample) and their average is 78, then x̄ = 78.
3.3. Step 3: Calculate the Standard Deviation (σ or s)
Calculate the standard deviation of the dataset. Use the population standard deviation (σ) if you have data for the entire population, and the sample standard deviation (s) if you are working with a sample.
The standard deviation measures the spread or variability of the data around the mean.
-
Population Standard Deviation (σ):
σ = √[(Σ(x – μ)^2) / N]
-
Sample Standard Deviation (s):
s = √[(Σ(x – x̄)^2) / (n – 1)]
For example, if the population standard deviation for the test scores is 10, then σ = 10. If the sample standard deviation is 8, then s = 8.
3.4. Step 4: Apply the Z Score Formula
Now that you have the raw score (x), the mean (μ or x̄), and the standard deviation (σ or s), you can calculate the Z score using the appropriate formula:
-
For Population:
z = (x – μ) / σ
-
For Sample:
z = (x – x̄) / s
Plug the values into the formula and perform the calculation.
For example, using the previous values:
- Raw score (x) = 85
- Population mean (μ) = 75
- Population standard deviation (σ) = 10
The Z score would be:
z = (85 – 75) / 10 = 1
This means the student’s test score is one standard deviation above the mean.
3.5. Step 5: Interpret the Z Score
The final step is to interpret the Z score. The Z score indicates how many standard deviations the raw score is from the mean.
- Positive Z Score: The raw score is above the mean.
- Negative Z Score: The raw score is below the mean.
- Z Score of Zero: The raw score is equal to the mean.
The magnitude of the Z score indicates how far the raw score is from the mean. For example:
- A Z score of 1 means the raw score is one standard deviation above the mean.
- A Z score of -2 means the raw score is two standard deviations below the mean.
Understanding the Z score helps you compare the raw score to the distribution of the data and assess its relative position.
4. Practical Examples of Z Score Calculation
To further illustrate how to calculate Z scores, let’s walk through a few practical examples. These examples will cover different scenarios and types of data.
4.1. Example 1: Test Scores
Suppose a student scores 90 on a math test. The class mean is 75, and the standard deviation is 10. Calculate the Z score for the student’s test score.
- Identify the Raw Score (x):
x = 90 - Determine the Mean (μ):
μ = 75 - Calculate the Standard Deviation (σ):
σ = 10 - Apply the Z Score Formula:
z = (x – μ) / σ
z = (90 – 75) / 10
z = 15 / 10
z = 1.5 - Interpret the Z Score:
The Z score is 1.5, which means the student’s test score is 1.5 standard deviations above the mean.
This indicates that the student performed significantly better than the average student in the class.
4.2. Example 2: Heights of Adults
The average height of adult males in a population is 5’10” (70 inches), with a standard deviation of 3 inches. What is the Z score for a man who is 6’2″ (74 inches) tall?
- Identify the Raw Score (x):
x = 74 inches - Determine the Mean (μ):
μ = 70 inches - Calculate the Standard Deviation (σ):
σ = 3 inches - Apply the Z Score Formula:
z = (x – μ) / σ
z = (74 – 70) / 3
z = 4 / 3
z ≈ 1.33 - Interpret the Z Score:
The Z score is approximately 1.33, meaning the man’s height is 1.33 standard deviations above the mean height of adult males.
This indicates that the man is taller than average but not an extreme outlier in terms of height.
4.3. Example 3: Product Weights
A manufacturing company produces widgets with a target weight of 50 grams. A sample of widgets is taken, and one widget weighs 48 grams. The sample mean weight is 51 grams, and the sample standard deviation is 2 grams. Calculate the Z score for this widget.
- Identify the Raw Score (x):
x = 48 grams - Determine the Sample Mean (x̄):
x̄ = 51 grams - Calculate the Sample Standard Deviation (s):
s = 2 grams - Apply the Z Score Formula:
z = (x – x̄) / s
z = (48 – 51) / 2
z = -3 / 2
z = -1.5 - Interpret the Z Score:
The Z score is -1.5, which means the widget’s weight is 1.5 standard deviations below the sample mean.
This indicates that the widget is lighter than average, and the manufacturing process may need to be adjusted to maintain consistent weights.
4.4. Example 4: Investment Returns
An investor’s portfolio has an average annual return of 12% over the past 10 years. The market average return (benchmark) is 10% with a standard deviation of 5%. Calculate the Z score for the investor’s portfolio return.
- Identify the Raw Score (x):
x = 12% - Determine the Mean (μ):
μ = 10% - Calculate the Standard Deviation (σ):
σ = 5% - Apply the Z Score Formula:
z = (x – μ) / σ
z = (12 – 10) / 5
z = 2 / 5
z = 0.4 - Interpret the Z Score:
The Z score is 0.4, meaning the investor’s portfolio return is 0.4 standard deviations above the market average.
This indicates that the investor’s portfolio performed slightly better than the market average.
These examples demonstrate the versatility of Z scores and their application across various domains. By following the step-by-step guide and practicing with different scenarios, you can confidently calculate and interpret Z scores in any context.
5. Using Z-Table to Find Probabilities
Once you have calculated the Z score, you can use a Z-table (also known as a standard normal table) to find the probability associated with that Z score. The Z-table provides the area under the standard normal curve to the left of the Z score, which represents the probability of a data point falling below that Z score.
5.1. Understanding Z-Table
A Z-table is a table that lists the cumulative probabilities corresponding to different Z scores. It is based on the standard normal distribution, which has a mean of 0 and a standard deviation of 1. The Z-table typically provides probabilities for Z scores ranging from -3.49 to 3.49.
The Z-table is structured as follows:
- The rows represent the Z score up to the first decimal place (e.g., 1.0, 1.1, 1.2).
- The columns represent the second decimal place of the Z score (e.g., 0.00, 0.01, 0.02).
- The values in the table represent the area under the standard normal curve to the left of the Z score, which is the cumulative probability.
5.2. Steps to Use Z-Table
To use the Z-table to find probabilities:
- Calculate the Z Score: First, calculate the Z score for the data point you are interested in, using the appropriate formula.
- Find the Z Score in the Table: Look up the Z score in the Z-table. Find the row corresponding to the first decimal place of the Z score and the column corresponding to the second decimal place.
- Read the Probability: The value at the intersection of the row and column is the cumulative probability associated with the Z score. This probability represents the area under the standard normal curve to the left of the Z score.
- Interpret the Probability: Interpret the probability in the context of your data. The probability tells you the likelihood of a data point falling below the given Z score.
5.3. Examples of Finding Probabilities Using Z-Table
Let’s go through a few examples to illustrate how to use the Z-table.
Example 1: Finding Probability for Z = 1.50
Suppose you have a Z score of 1.50. To find the probability associated with this Z score:
- Locate 1.5 in the Row: Find the row labeled “1.5” in the Z-table.
- Locate 0.00 in the Column: Find the column labeled “0.00” (since the Z score is 1.50, the second decimal place is 0).
- Read the Probability: The value at the intersection of row 1.5 and column 0.00 is 0.9332.
This means that the probability of a data point falling below a Z score of 1.50 is 0.9332, or 93.32%.
Example 2: Finding Probability for Z = -0.75
Suppose you have a Z score of -0.75. To find the probability associated with this Z score:
- Locate -0.7 in the Row: Find the row labeled “-0.7” in the Z-table.
- Locate 0.05 in the Column: Find the column labeled “0.05” (since the Z score is -0.75, the second decimal place is 5).
- Read the Probability: The value at the intersection of row -0.7 and column 0.05 is 0.2266.
This means that the probability of a data point falling below a Z score of -0.75 is 0.2266, or 22.66%.
Example 3: Finding Probability Between Two Z Scores
Suppose you want to find the probability of a data point falling between two Z scores, Z1 = -0.50 and Z2 = 1.00.
- Find the Probability for Z1 = -0.50:
- Locate -0.5 in the row and 0.00 in the column.
- The probability is 0.3085.
- Find the Probability for Z2 = 1.00:
- Locate 1.0 in the row and 0.00 in the column.
- The probability is 0.8413.
- Calculate the Difference:
- Subtract the probability for Z1 from the probability for Z2:
-
- 8413 – 0.3085 = 0.5328
This means that the probability of a data point falling between a Z score of -0.50 and 1.00 is 0.5328, or 53.28%.
5.4. Using Z-Table for Right-Tail Probabilities
Sometimes, you may need to find the probability of a data point falling above a certain Z score (right-tail probability). To do this:
-
Find the Probability for the Z Score: Look up the Z score in the Z-table and find the cumulative probability.
-
Subtract from 1: Subtract the cumulative probability from 1 to get the right-tail probability.
P(Z > z) = 1 – P(Z < z)
For example, if you want to find the probability of a data point falling above a Z score of 1.50:
- Find the Probability for Z = 1.50:
- From the Z-table, the probability is 0.9332.
- Subtract from 1:
- 1 – 0.9332 = 0.0668
This means that the probability of a data point falling above a Z score of 1.50 is 0.0668, or 6.68%.
6. Common Mistakes and How to Avoid Them
Calculating Z scores and using Z-tables can be prone to errors if not done carefully. This section highlights common mistakes and provides guidance on how to avoid them.
6.1. Using the Wrong Formula
Mistake: Using the population formula when you should be using the sample formula, or vice versa.
How to Avoid: Always double-check whether you are working with the entire population or just a sample. If you have data for the entire population, use the population mean (μ) and standard deviation (σ). If you have data for a sample, use the sample mean (x̄) and standard deviation (s). Remember that the sample standard deviation formula has (n – 1) in the denominator.
6.2. Incorrectly Calculating Mean and Standard Deviation
Mistake: Making errors when calculating the mean or standard deviation.
How to Avoid: Take your time and double-check your calculations. Use a calculator or statistical software to minimize errors. Ensure you are using the correct formulas for mean and standard deviation. For the standard deviation, be careful with the squaring and square root operations.
6.3. Misinterpreting Z-Table Values
Mistake: Misunderstanding what the values in the Z-table represent.
How to Avoid: Remember that the Z-table provides the cumulative probability to the left of the Z score. If you need to find the probability to the right, subtract the Z-table value from 1. If you need to find the probability between two Z scores, subtract the smaller probability from the larger probability.
6.4. Confusing Positive and Negative Z Scores
Mistake: Getting confused with positive and negative Z scores when looking up values in the Z-table.
How to Avoid: Pay close attention to the sign of the Z score. Use the appropriate section of the Z-table for positive and negative values. A negative Z score indicates that the data point is below the mean, while a positive Z score indicates that it is above the mean.
6.5. Rounding Errors
Mistake: Rounding intermediate calculations too early, which can lead to inaccurate final Z scores and probabilities.
How to Avoid: Keep as many decimal places as possible during the intermediate calculations. Only round the final Z score and probability to the desired number of decimal places.
6.6. Not Checking for Normality
Mistake: Applying Z scores and Z-tables to data that is not normally distributed.
How to Avoid: Check whether your data is approximately normally distributed before using Z scores. You can use histograms, normal probability plots, or statistical tests to assess normality. If the data is not normally distributed, consider using other statistical methods that do not rely on this assumption.
6.7. Ignoring Outliers
Mistake: Not addressing outliers in the data, which can significantly affect the mean and standard deviation, and thus the Z scores.
How to Avoid: Identify and examine outliers in your dataset. Determine whether they are genuine data points or errors. If they are errors, correct or remove them. If they are genuine data points, consider using robust statistical methods that are less sensitive to outliers.
6.8. Using Inappropriate Z-Table
Mistake: Using a Z-table that is not standard or is designed for a specific purpose (e.g., one-tailed tests).
How to Avoid: Ensure you are using a standard Z-table that provides cumulative probabilities from the left. If you are conducting a one-tailed test, adjust your calculations accordingly. Always read the instructions and headings on the Z-table to understand what values it provides.
By being aware of these common mistakes and following the recommended steps to avoid them, you can improve the accuracy and reliability of your Z score calculations and interpretations.
7. Advanced Applications of Z Score
While the basic calculation and interpretation of Z scores are essential, the Z score also has several advanced applications that are valuable in more complex statistical analyses.
7.1. Hypothesis Testing
Z scores are used in hypothesis testing to determine whether a sample mean is significantly different from a population mean. This is a fundamental aspect of statistical inference.
The steps for hypothesis testing using Z scores are:
- State the Hypotheses:
- Null Hypothesis (H0): There is no significant difference between the sample mean and the population mean.
- Alternative Hypothesis (H1): There is a significant difference between the sample mean and the population mean.
- Set the Significance Level (α):
- Choose a significance level (e.g., α = 0.05), which represents the probability of rejecting the null hypothesis when it is true.
- Calculate the Z Score:
- Use the Z score formula to calculate the Z score for the sample mean.
- Determine the Critical Value:
- Find the critical value(s) from the Z-table based on the significance level and the type of test (one-tailed or two-tailed).
- Make a Decision:
- Compare the calculated Z score to the critical value(s).
- If the absolute value of the Z score is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
For example, if you are testing whether a new teaching method improves test scores, you would compare the mean test score of students taught using the new method to the population mean of students taught using the traditional method.
7.2. Confidence Intervals
Z scores are used to construct confidence intervals, which provide a range of values within which the population mean is likely to fall.
The formula for calculating a confidence interval using Z scores is:
Confidence Interval = x̄ ± (Zα/2 * (σ / √n))
Where:
- x̄ is the sample mean
- Zα/2 is the critical value from the Z-table for the desired confidence level (e.g., 1.96 for a 95% confidence interval)
- σ is the population standard deviation
- n is the sample size
For example, if you want to estimate the average height of adult females with a 95% confidence interval, you would use the sample mean height, the population standard deviation, and the Z score corresponding to a 95% confidence level (1.96) to calculate the interval.
7.3. Process Control
In manufacturing and other industries, Z scores are used to monitor process control. By calculating Z scores for key metrics, companies can identify deviations from expected values and take corrective actions to maintain product quality.
Control charts, such as X-bar charts and S charts, use Z scores to track process performance over time. Data points that fall outside the control limits (typically ±3 standard deviations from the mean) indicate that the process is out of control and requires investigation.
7.4. Comparing Scores on Different Scales
Z scores are used to compare scores on different scales. By standardizing the scores, you can directly compare performance across different tests or metrics.
For example, if a student takes two tests, one with a mean of 70 and a standard deviation of 10, and another with a mean of 80 and a standard deviation of 5, you can calculate the Z scores for each test to compare the student’s relative performance.
7.5. Prediction Intervals
Z scores can be used to calculate prediction intervals, which provide a range of values within which a future observation is likely to fall.
The formula for calculating a prediction interval is:
Prediction Interval = x̄ ± (Zα/2 s √(1 + (1/n)))
Where:
- x̄ is the sample mean
- Zα/2 is the critical value from the Z-table for the desired confidence level
- s is the sample standard deviation
- n is the sample size
For example, if you want to predict the weight of the next widget produced by a manufacturing process, you would use the sample mean weight, the sample standard deviation, and the Z score corresponding to the desired confidence level to calculate the prediction interval.
7.6. Six Sigma
Six Sigma is a quality management methodology that aims to reduce defects and improve process performance. Z scores play a key role in Six Sigma by quantifying the capability of a process.
The Z score, in this context, represents the number of standard deviations between the process mean and the nearest specification limit. A higher Z score indicates a more capable process with fewer defects. A process with a Z score of 6 (Six Sigma) has a very low defect rate (3.4 defects per million opportunities).
These advanced applications demonstrate the versatility and power of Z scores in statistical analysis. By mastering these techniques, you can gain deeper insights from your data and make more informed decisions.
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9. Frequently Asked Questions (FAQs) About Z Scores
To provide further clarity on Z scores, here are some frequently asked questions along with detailed answers.
9.1. What is the difference between a Z score and a T score?
A Z score is used when you know the population standard deviation or have a large sample size (typically n > 30). A T score is used when you do not know the population standard deviation and have a small sample size (typically n < 30). T scores account for the uncertainty introduced by estimating the standard deviation from a small sample.
9.2. How do I interpret a Z score of 0?
A Z score of 0 indicates that the data point is exactly at the mean of the distribution. It means that the value is neither above nor below the average.
9.3. What is a good Z score?
The interpretation of a “good” Z score depends on the context. In some cases, a Z score close to 0 is desirable, indicating that the value is close to the average. In other cases, a large positive or negative Z score may be desirable, indicating that the value is significantly above or below the average. Generally, Z scores greater than 2 or less than -2 are considered significant.
9.4. Can a Z score be greater than 3 or less than -3?
Yes, Z scores can be greater than 3 or less than -3. These values indicate that the data point is more than 3 standard deviations away from the mean, which is relatively rare in a normal distribution. Data points with Z scores in this range are often considered outliers.
9.5. How do I calculate the Z score for a sample?
To calculate the Z score for a sample, use the formula:
z = (x – x̄) / s
Where:
- x is the raw data point
- x̄ is the sample mean
- s is the sample standard deviation
9.6. What does the Z-table tell me?
The Z-table (or standard normal table) provides the cumulative probability associated with a given Z score. The probability represents the area under the standard normal curve to the left of the Z score, which is the likelihood of a data point falling below that Z score.
9.7. How do I find the area to the right of a Z score using the Z-table?
To find the area to the right of a Z score, subtract the Z-table value (cumulative probability to the left) from 1.
Area to the right = 1 – P(Z < z)
9.8. How do I use Z scores in hypothesis testing?
In hypothesis testing, Z scores are used to determine whether a sample mean is significantly different from a population mean. You calculate the Z score for the sample mean and compare it to a critical value from the Z-table. If the absolute value of the Z score is greater than the critical value, you reject the null hypothesis.
9.9. What are the assumptions for using Z scores?
The main assumption for using Z scores is that the data is approximately normally distributed. Additionally, you should have a large enough sample size or know the population standard deviation.
9.10. Can Z scores be used for non-normal distributions?
While Z scores are based on the assumption of normality, they can still be used for non-normal distributions
