Enter a list of numbers to calculate mean, median, mode, range, quartiles, IQR, variance, standard deviation, five-number summary, outliers, geometric mean, and more. See formulas, visual distributions, and step-by-step explanations.
Statistics Calculator
Enter your data below and click Calculate for a complete data summary.
Separate values with commas, spaces, semicolons, or line breaks, whichever is easiest. Positive, negative, decimal, and zero values are all fine.
Exclusive (Median-of-Halves / Tukey-style)
Mean (Average)
0
n = 0
Sample Standard Deviation
0
Median = 0
Your Data Summary
Five-Number Summary — Box-and-Whisker Plot
Where Your Values Fall
Variance & Standard Deviation
Find a Z-Score for a Value
Enter any value to see how many standard deviations it sits from the mean, using your selected Sample or Population setting.
Descriptive statistics are numbers that summarize and describe a dataset. Suppose you have 8 test scores: 50, 62, 68, 72, 72, 75, 80, 91. Looking at the raw list gives you information, but it may not immediately answer questions such as: What is the typical score? What is the middle score? Which score occurs most often? How far apart are the scores? Are most values clustered together? Is there an unusually high or low value?
Statistics such as mean, median, mode, range, quartiles, interquartile range, variance, and standard deviation help answer those questions. Descriptive statistics do not automatically tell you why the data looks the way it does. They summarize what is in the data you entered. That is the purpose of this Statistics Calculator.
Statistics Turn Raw Numbers Into a Data Summary
What Is the Mean?
The arithmetic mean is what most people call the average. Formula: Mean = Sum of Values ÷ Number of Values. Using 2, 4, 4, 6, 9: first add the values, 2 + 4 + 4 + 6 + 9 = 25. There are 5 values. Mean: 25 ÷ 5 = 5.
For a sample, the mean is often written x̄. For a population, it is commonly written μ. The mean uses every value in the dataset. That makes it informative, but it also means an extreme observation can move it noticeably.
The Mean Uses Every Value
Every dot contributes to where the mean lands, like weights balanced along a number line.
What Is the Median?
The median is the middle value after the data is sorted. For 2, 4, 4, 6, 9, there are 5 values. The middle value is 4. So the median is 4. If there are an even number of observations, average the two middle values. Example: 2, 4, 6, 10. Middle values: 4 and 6. Median: (4 + 6) ÷ 2 = 5.
The median depends on position rather than the size of every value. That makes it more resistant to extreme values than the mean.
Mean or Median: Which Is Better?
Neither measure is always better. The appropriate choice depends on the data. Consider 10, 11, 12, 13, 50. Median: 12. Mean: (10 + 11 + 12 + 13 + 50) ÷ 5 = 96 ÷ 5 = 19.2. The very high value, 50, pulls the mean upward. The median remains 12 because it is still the middle observation. For strongly skewed data or data containing extreme values, the median can sometimes describe the typical center better. For relatively symmetric data without strong outliers, the mean is often useful because it incorporates every observation.
Extreme Values Can Move the Mean
Data: 10, 11, 12, 13, 50. Median = 12. Mean = 19.2. The high value of 50 pulls the mean to the right, while the median stays near the cluster.
What Is the Mode?
The mode is the most frequently occurring value. Example: 2, 4, 4, 6, 9. The value 4 appears twice. The others appear once. Mode: 4. A dataset can have one mode, multiple modes, or no mode. Example: 1, 1, 2, 2, 3 has two modes, 1 and 2. If every value occurs only once, there is no mode. Mode is especially useful when the most common value matters. Unlike the mean and median, mode can also be useful for some categorical data.
What Is the Range?
Range measures the distance from the smallest value to the largest value. Formula: Range = Maximum − Minimum. For 2, 4, 4, 6, 9: Maximum: 9. Minimum: 2. Range: 9 − 2 = 7. Range is very easy to understand. But it depends entirely on only two values. That means one unusually high or low observation can dramatically change it.
Range
Min = 2, Max = 9, Range = 7. The bracket spans the full distance between the smallest and largest values.
What Are Quartiles?
Quartiles divide ordered data into portions. The main quartiles are Q1, the lower quartile; Q2, the median; and Q3, the upper quartile. Together with minimum and maximum, they create the five-number summary. The five-number summary is Minimum, Q1, Median, Q3, Maximum. This is one of the most useful compact descriptions of a dataset's position and spread.
Quartile values can differ slightly depending on the statistical convention or software used. That is why a responsible calculator should identify which quartile method it uses rather than presenting every possible method as though they must always produce identical answers.
Five-Number Summary
Minimum
0%
→
Q1
25%
→
Median
50%
→
Q3
75%
→
Maximum
100%
These positions describe the ordered dataset from smallest to largest.
What Is the Interquartile Range?
The interquartile range, or IQR, measures the spread of the middle portion of the dataset. Formula: IQR = Q3 − Q1. Because the IQR uses the quartiles rather than only the minimum and maximum, it is less influenced by extremely low or high values than the full range. That makes IQR especially useful when data is skewed or contains potential outliers.
How Are Potential Outliers Found Using IQR?
One widely used rule uses 1.5 × IQR. Calculate Lower Fence = Q1 − 1.5(IQR). Upper Fence = Q3 + 1.5(IQR). Any observation below the lower fence or above the upper fence can be flagged as a potential outlier.
The word potential matters. A value outside the fences is not automatically an error. It could represent a legitimate unusual observation, a data-entry problem, a rare event, or an important part of the data. Statistics can flag it. Context determines what it means.
Box Plot with a Potential Outlier
Data: 2, 3, 4, 5, 6, 7, 30. Q1 = 3, Median = 5, Q3 = 7, IQR = 4. Upper fence = 7 + 1.5(4) = 13. 30 lies beyond the upper fence and is marked as a potential outlier.
What Is Variance?
Variance measures how far values tend to spread away from the mean using squared deviations. The basic idea: find the mean, subtract the mean from each observation, square each deviation, add those squared deviations, then divide using the appropriate population or sample denominator.
Why square the deviations? If we simply added positive and negative deviations from the mean, they would tend to cancel each other. Squaring makes every deviation nonnegative and gives greater influence to observations farther from the mean. Population variance is commonly written σ². Sample variance: s².
What Is Standard Deviation?
Standard deviation is the square root of variance. Population standard deviation: σ. Sample standard deviation: s. Standard deviation returns the measure of spread to the original units of the data. If test scores are measured in points, variance is measured in points², while standard deviation is again measured in points. That makes standard deviation easier to interpret. If you want a deeper calculation showing the deviation of every value, use the dedicated Standard Deviation Calculator.
Smaller vs Larger Standard Deviation
Dataset A clusters tightly around its mean (smaller standard deviation). Dataset B is more spread out around the same center (larger standard deviation).
What Is the Difference Between Sample and Population Standard Deviation?
A population contains every observation in the group you want to describe. A sample contains only part of a larger population. Population variance uses N in the denominator. Sample variance commonly uses n − 1. That adjustment helps compensate for the fact that a sample is being used to estimate population variability.
For this reason, it is important to know whether your data represents the entire group or a subset of the group. This calculator shows both values while allowing you to specify which one is most relevant.
What Is the Geometric Mean?
The geometric mean is another measure of center. Instead of adding values and dividing, the geometric mean is based on multiplication. Formula: GM = (x₁x₂...xₙ)^(1/n). Example: 1, 4, 16. Product: 1 × 4 × 16 = 64. Cube root: ∛64 = 4.
The geometric mean can be useful when quantities relate through multiplicative change, growth factors, ratios, or compounded changes. It is not a replacement for the arithmetic mean in every dataset. Different averages answer different questions.
Arithmetic Mean vs Geometric Mean
Arithmetic Mean
↓
Add: 1 + 4 + 16 = 21
↓
Divide by 3: 7
Geometric Mean
↓
Multiply: 1 × 4 × 16 = 64
↓
3rd Root of 64: 4
Same data, 1, 4, 16, two different questions. Arithmetic mean adds then divides. Geometric mean multiplies then takes the nth root.
What Is the Harmonic Mean?
The harmonic mean is another type of average. Formula: HM = n ÷ Σ(1/x). It is often useful for certain types of rates, ratios, and reciprocal relationships. Example: 2, 3, 6. Reciprocals: 1/2 + 1/3 + 1/6. Using a common denominator: 3/6 + 2/6 + 1/6 = 1. There are 3 values. Harmonic Mean: 3 ÷ 1 = 3. Like the geometric mean, the harmonic mean has specific uses. It should not be selected simply because it produces a different number.
What Is the Five-Number Summary Used For?
The five-number summary provides a quick picture of center, spread, and possible skew. It is especially useful because it forms the basis of the box-and-whisker plot. A box plot makes it easier to compare median, quartile spread, overall spread, and potential outliers. This can reveal patterns that a single mean alone would not show.
Box Plot Anatomy
The box spans Q1 to Q3, the middle 50% of the data. Its width equals the IQR. The line inside the box marks the median.
What Is a Z-Score?
A z-score tells you how many standard deviations a value is above or below the mean. For a population: z = (x − μ) ÷ σ. Example: Mean: 50. Standard Deviation: 10. Observed value: 70. Then z = (70 − 50) ÷ 10 = 20 ÷ 10 = 2. The value 70 is two standard deviations above the mean. If z = −1.5, the value is 1.5 standard deviations below the mean. A z-score describes relative position. It does not automatically tell you whether a value is good, bad, normal, or abnormal. That depends on the context.
Z-Score Number Line
Mean = 50, Standard Deviation = 10, Value = 70. z = 2, two standard deviations above the mean.
What Is a Frequency Distribution?
Frequency tells you how often each value occurs. Example: 1, 1, 2, 3, 3, 3. Frequencies: 1 appears 2 times. 2 appears 1 time. 3 appears 3 times. Relative frequency converts those counts into proportions or percentages. There are 6 total observations. Relative frequency of 3 is 3 ÷ 6 = 50%. Cumulative frequency keeps a running total as values increase. Frequency tables are especially helpful when the same values occur repeatedly.
Frequency Table Example
Value
Frequency
Relative Frequency
1
2
33.33%
2
1
16.67%
3
3
50%
What Is a Percentile?
A percentile describes a value's relative position within an ordered set or distribution. For example, saying a result is near the 75th percentile generally means it is at or above a large portion of the observations according to the percentile method being used. Percentile conventions can vary, especially for small datasets. That is another reason calculators should state their method. Do not assume a percentile and a percentage score mean the same thing. A score of 80% and 80th percentile describe different ideas.
What Is the Empirical Rule?
For data that is reasonably modeled by a normal distribution, the 68-95-99.7 Rule provides a useful approximation. Approximately 68% of values lie within 1 standard deviation of the mean. Approximately 95% within 2. Approximately 99.7% within 3. This is a property of the normal distribution. It should NOT be blindly applied to every dataset. A dataset can be skewed, multi-peaked, irregular, or contain outliers. The rule is most meaningful when a normal model is appropriate.
The Empirical Rule — Approximation for Normal Distributions
Approximation for normal distributions. About 68% of values fall within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean.
Another Statistics Formula Reference
East Central College's Learning Center provides a statistics equations and formulas reference that includes measures of center, measures of dispersion, range, IQR, the five-number summary, outlier fences, z-scores, percentiles, frequency concepts, and sample-versus-population notation. For another educational formula reference, see their statistics equations and formulas guide. This is an external educational cross-reference only; CalculateThisWay's article, examples, and diagrams are original.
How Do You Read a Dataset as a Whole?
Do not rely on one statistic alone. Suppose two datasets have the same mean. They can still have very different spread, shape, outliers, and quartiles. A useful first look often includes: count, mean, median, minimum and maximum, range, quartiles, IQR, standard deviation, visual distribution, and potential outliers. Together these tell a much richer story than "the average is 50."
When Should You Use Mean, Median, or Mode?
Use the mean when every numerical value should contribute to the center and strong extreme values are not making it misleading. Use the median when the middle position is important or the data is skewed or contains extreme observations. Use the mode when the most common value or category matters. It is common for analysts to look at more than one measure. If mean and median are very different, that may itself tell you something about the shape of the data.
What Does a Large IQR Mean?
A larger IQR means the middle half of the data is spread over a wider interval. A smaller IQR means the middle half is more concentrated. But whether an IQR is large or small depends on the scale and subject. An IQR of 10 may be large for one type of measurement and tiny for another. Statistics should always be interpreted with units and context.
What Does a Large Standard Deviation Mean?
A larger standard deviation means observations tend to be farther from the mean. A smaller standard deviation means they tend to cluster more closely around the mean. Again, large and small depend on the scale. A standard deviation should not be interpreted without understanding what the numbers measure.
Can Two Datasets Have the Same Mean but Different Spread?
Yes. Dataset A: 48, 49, 50, 51, 52. Mean: 50. Dataset B: 10, 30, 50, 70, 90. Mean: 50. The means are identical. But Dataset B is much more spread out. This demonstrates why mean alone cannot fully describe a dataset.
Same Center. Very Different Spread.
Same mean, very different spread. Both datasets average to 50, but Dataset B ranges from 10 to 90 while Dataset A stays within 48 to 52.
Where Are Descriptive Statistics Used?
Descriptive statistics appear almost anywhere numerical data is collected.
Education: test scores, grades, attendance, class performance.
Business: sales, customer behavior, production metrics, inventory.
Quality control: manufacturing measurements and process data.
The numbers themselves vary by field. The basic statistical questions remain similar: Where is the center? How much spread is there? Are there unusual values? What does the distribution look like?
What Is the Difference Between Descriptive and Inferential Statistics?
Descriptive statistics summarize the data you actually observed. This calculator is primarily a descriptive statistics tool. Inferential statistics uses sample data to draw conclusions or estimate characteristics of a larger population. Inferential methods include concepts such as confidence intervals, hypothesis tests, and statistical models. A descriptive calculator should not pretend that a mean or standard deviation alone proves something about an entire population.
What Are the Most Common Statistics Mistakes?
Using the mean without checking outliers. A strong extreme observation can move the mean.
Forgetting to sort before finding the median or quartiles. Position statistics depend on ordered data.
Assuming every dataset has one mode. Some datasets have multiple modes or no mode.
Confusing range with IQR. Range uses the extremes. IQR measures the middle 50%.
Confusing sample and population standard deviation. The formulas use different denominators.
Assuming an outlier is an error. Potential outliers may be legitimate data.
Comparing standard deviations without context. Units and scale matter.
Using the empirical rule on non-normal data. The 68-95-99.7 Rule assumes an approximately normal model.
Confusing percentile with percent. They measure different things.
Rounding too early. Keep full internal precision and round only displayed results.
What Is a Good Step-by-Step Method for Descriptive Statistics?
Check the data. Make sure values are entered correctly.
Sort the values. Sorting helps reveal minimum, maximum, median, quartiles, and repeated values.
Find the center. Calculate mean, median, mode.
Find the spread. Calculate range, IQR, variance, standard deviation.
Build the five-number summary. Minimum, Q1, Median, Q3, Maximum.
Check for potential outliers. Use the 1.5 × IQR rule when appropriate.
Look at the distribution. Use a dot plot, histogram, or box plot.
Interpret the results together. Do not rely on a single number.
What Should You Leave This Page Knowing?
Do not leave knowing only "my mean is 55." Leave knowing: I understand what descriptive statistics do. I know how the mean is calculated. I understand how the median differs from the mean. I know what a mode represents. I understand range. I know what quartiles are. I understand the five-number summary. I know what IQR measures. I understand the 1.5 × IQR potential-outlier rule. I know the difference between variance and standard deviation. I understand the difference between sample and population standard deviation. I know what geometric mean is. I understand harmonic mean has specialized uses. I know what a z-score represents. I understand frequency and relative frequency. I understand that percentile methods can vary. I know the 68-95-99.7 Rule applies to an appropriate normal model. I know two datasets can have the same mean and very different spread. And most importantly, I can look at the results together and understand what my data is showing.
That is what the CalculateThisWay Statistics Calculator should help someone do.
Statistics Formulas at a Glance
Statistic
Formula
Count
n
Sum
Σx
Sample Mean
x̄ = Σx ÷ n
Population Mean
μ = Σx ÷ N
Range
Maximum − Minimum
IQR
Q3 − Q1
Lower Outlier Fence
Q1 − 1.5(IQR)
Upper Outlier Fence
Q3 + 1.5(IQR)
Population Variance
σ² = Σ(x − μ)² ÷ N
Population SD
σ = √σ²
Sample Variance
s² = Σ(x − x̄)² ÷ (n − 1)
Sample SD
s = √s²
Geometric Mean
(Πxᵢ)^(1/n)
Harmonic Mean
n ÷ Σ(1/xᵢ)
Midrange
(Minimum + Maximum) ÷ 2
Population z-score
z = (x − μ) ÷ σ
Worked Examples
Example 1: Mean. Data: 2, 4, 4, 6, 9.
Sum: 25. Count: 5.
Mean: 25 ÷ 5 = 5
Example 2: Median. Data: 2, 4, 4, 6, 9.
Median: 4
Example 3: Even Median. Data: 2, 4, 6, 10.
Middle values: 4 and 6.
Median: (4 + 6) ÷ 2 = 5
Example 4: Mode. Data: 2, 4, 4, 6, 9.
Mode: 4
Example 5: Range. Data: 2, 4, 4, 6, 9.
Range: 9 − 2 = 7
Example 6: Geometric Mean. Data: 1, 4, 16.
Product: 1 × 4 × 16 = 64.
Geometric Mean: ∛64 = 4
Example 7: Harmonic Mean. Data: 2, 3, 6.
1/2 + 1/3 + 1/6 = 1. There are 3 values.
Harmonic Mean: 3 ÷ 1 = 3
Example 8: Z-Score. Mean = 50, SD = 10, Value = 70.
z = (70 − 50) ÷ 10
z = 2
Example 9: Mean vs Median with an Outlier. Data: 10, 11, 12, 13, 50.
Mean: 96 ÷ 5 = 19.2. Median: 12.
Mean = 19.2, Median = 12
Example 10: Frequency. Data: 1, 1, 2, 3, 3, 3.
Frequency of 3 is 3 out of 6 values.
Relative Frequency of 3: 50%
Try It Yourself
Data: 3, 5, 7, 7, 8. What is the mean?
Questions People Actually Ask
What is a statistics calculator?
A tool that takes a list of numbers you enter and computes descriptive statistics such as mean, median, mode, range, quartiles, IQR, variance, standard deviation, and more.
What is descriptive statistics?
Numbers that summarize and describe a data set, such as its center, spread, and shape, rather than drawing conclusions about a larger population.
How do I calculate the mean?
Add every value together, then divide that sum by the number of values.
What is the difference between mean and average?
In everyday use they mean the same thing. "Average" most commonly refers to the arithmetic mean, though statisticians also use median or mode as other measures of "average."
How do I calculate the median?
Sort the values from smallest to largest. If there is an odd number of values, the median is the middle one. If there is an even number, average the two middle values.
How do I find the median with an even number of values?
Sort the data, find the two middle values, and average them.
What is the mode?
The value that occurs most frequently in a data set.
Can a dataset have two modes?
Yes. When two values are tied for the highest frequency, the data set is called bimodal and both values are reported as modes.
Can a dataset have no mode?
Yes. If every value occurs the same number of times, there is no single most frequent value, so the data set has no mode.
What is range in statistics?
The distance between the maximum and minimum values: Range = Maximum minus Minimum.
How do I find Q1 and Q3?
Sort the data, split it into a lower half and an upper half (excluding the overall median for an odd-sized data set), then Q1 is the median of the lower half and Q3 is the median of the upper half.
What is a quartile?
A value that divides ordered data into quarters; Q1, Q2 (the median), and Q3 mark the 25th, 50th, and 75th positions.
Why do quartile calculators sometimes give different answers?
Different statistical conventions calculate quartiles slightly differently. This calculator uses the exclusive, median-of-halves (Tukey-style) method and states that clearly in the results.
What is the five-number summary?
Minimum, Q1, Median, Q3, and Maximum: a compact description of a data set's center, spread, and shape.
What is IQR?
The interquartile range, IQR = Q3 minus Q1, which measures the spread of the middle 50% of the data.
How do I calculate IQR?
Find Q1 and Q3, then subtract: IQR = Q3 - Q1.
How do I identify potential outliers?
Compute the lower fence (Q1 - 1.5 x IQR) and upper fence (Q3 + 1.5 x IQR). Any value below the lower fence or above the upper fence can be flagged as a potential outlier.
What does the 1.5 IQR rule mean?
It is a widely used rule that flags values more than 1.5 times the IQR beyond Q1 or Q3 as potential outliers worth a closer look.
Does an outlier mean the data is wrong?
No. A potential outlier may be a legitimate observation. Statistical context should determine whether it is retained, investigated, or excluded.
What is variance?
The average of the squared deviations from the mean, using the appropriate population or sample denominator.
What is standard deviation?
The square root of variance; it measures spread in the same units as the original data.
What is the difference between variance and standard deviation?
Variance is the average squared deviation, in squared units. Standard deviation is the square root of variance, returning to the original units.
What is population variance?
Variance calculated when your data represents the entire group you want to describe; it divides by N.
What is sample variance?
Variance calculated when your data is a subset of a larger group; it divides by n-1, the common sample variance correction.
What is population standard deviation?
The square root of population variance, describing spread across an entire population.
What is sample standard deviation?
The square root of sample variance, estimating spread in a population from a sample.
What is the difference between sample and population standard deviation?
Population standard deviation divides by N; sample standard deviation divides by n-1 to correct for using a subset to estimate the whole.
What is geometric mean?
A measure of center based on multiplying values together and taking the nth root, useful for growth rates and multiplicative relationships. It requires strictly positive values.
When should I use geometric mean?
When quantities relate through multiplicative change, growth factors, ratios, or compounded changes, rather than simple addition.
What is harmonic mean?
An average based on the reciprocals of the values, often useful for certain rates and ratios.
What is midrange?
The average of the minimum and maximum values. It is easy to compute but highly sensitive to extreme values.
What is a z-score?
A number telling you how many standard deviations a value is above or below the mean.
How do I calculate a z-score?
Subtract the mean from the value, then divide by the standard deviation: z = (x - mean) / SD.
What does a z-score of 2 mean?
The value is 2 standard deviations above the mean.
What is a frequency distribution?
A summary showing how often each value occurs in a data set.
What is relative frequency?
A value's frequency expressed as a proportion or percentage of the total number of observations.
What is cumulative frequency?
A running total of frequencies as values increase through the sorted data.
What is a percentile?
A value describing relative position within an ordered set; percentile conventions can vary, especially for small data sets.
Is percentile the same as percent?
No. An 80% score and an 80th percentile describe different ideas; they should not be assumed to mean the same thing.
What is the 68-95-99.7 rule?
For approximately normal data, about 68% of values fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.
Can two datasets have the same mean but different standard deviations?
Yes. Identical means can come from very differently spread-out data, which is exactly what standard deviation measures.
Is mean or median better when there are outliers?
The median is often more representative of the typical value when strong outliers or skew are present, because it depends on position rather than magnitude.
What statistics should I look at first?
Count, mean, median, minimum and maximum, range, quartiles, IQR, standard deviation, a visual distribution, and any potential outliers together give a much richer picture than any single number.
Sources & Further Reading
East Central College Learning Center: Statistics Equations & Formulas.statistics equations and formulas reference. Referenced as an external educational cross-reference for foundational statistics concepts. CalculateThisWay's article, examples, and diagrams are original and are not copied from this or any other source.
Statistics Calculator or Another Math Tool?
Statistics Calculator
I have a list of numbers and want a complete descriptive summary including center, spread, quartiles, outliers, and visualizations.
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Standard Deviation Calculator
I specifically need variance or standard deviation with detailed deviation calculations.
Data is parsed from the values you enter and sorted numerically before any positional statistic (median, quartiles, five-number summary) is calculated. Mean uses the ordinary arithmetic average. Population variance divides the sum of squared deviations by N; sample variance divides by n − 1, the common sample variance correction. Standard deviation is calculated as the square root of full-precision variance, never from an already-rounded standard deviation. Quartiles (Q1 and Q3) use the Exclusive method (median-of-halves / Tukey-style): for an odd-sized data set, the overall median is excluded before splitting the sorted values into a lower half and an upper half; Q1 is the median of the lower half and Q3 is the median of the upper half. The same method is used consistently for both the quartiles shown in your results and the 1.5 × IQR potential-outlier fences, so the two never mix conventions. Decimals may be rounded for display while calculations use greater internal precision. Geometric mean is shown only for strictly positive data; harmonic mean is shown only when no value is zero.
Last reviewed: September 2, 2026
This calculator is provided for general-purpose statistical analysis, education, and research support. Descriptive statistics summarize the data you entered; they do not by themselves prove anything about a larger population, and a potential outlier is not automatically an error. Quartile conventions vary between statistical software, and the empirical rule applies specifically to approximately normal distributions. Always double-check results that will be used for graded coursework, professional work, or any other high-stakes purpose.