Z-Score Calculator
Get the z-score of any value from the mean and standard deviation, with its percentile and probabilities, or paste a data set to z-score every value.
- Z-score and percentile
- Left, right and two-tailed
- Z-scores for a data set
- Steps shown
Z-score of one value
Mean and SD come from the data set below (sample SD). Type your own to unlink.
z-score
1.5435
- Percentile
- 93.86%
- P(Z < z)
- 0.9386
- P(Z > z)
- 0.0614
- P(−|z| < Z < |z|)
- 0.8773
- Two-tailed p
- 0.1227
Data set (optional)
10 values.
Results
Mean (x̄)
1,160
Standard deviation (s)
97.1825
Count (n)
10
Median
1,175
Center
- Count (n)
- 10
- Sum (Σx)
- 11,600
- Mean
- 1,160
- Median
- 1,175
- Mode
- No mode
Spread
- Sample standard deviation (s)
- 97.1825
- Population standard deviation (σ)
- 92.1954
- Sample variance (s²)
- 9,444.4444
- Population variance (σ²)
- 8,500
- Sum of squares (SS)
- 85,000
- Mean absolute deviation
- 76
- Standard error of the mean
- 30.7318
- Coefficient of variation (s ÷ mean)
- 8.38%
- Minimum
- 990
- Maximum
- 1,310
- Range
- 320
Quartiles and outliers
- Q1
- 1,090
- Q3
- 1,220
- Interquartile range (IQR)
- 130
- Five number summary
- 990, 1,090, 1,175, 1,220, 1,310
- Outlier fences
- 895 and 1,415
- Outliers (1.5 × IQR)
- None
Empirical rule (using s)
- Within 1 SD (68% if normal)
- 6 of 10 (60%)
- Within 2 SD (95% if normal)
- 10 of 10 (100%)
- Within 3 SD (99.7% if normal)
- 10 of 10 (100%)
Step-by-step working
Find the mean of the data
x̄ = Σx ÷ n
x̄ = 11,600 ÷ 10 = 1,160
Find the sample standard deviation
s = √(SS ÷ (n − 1))
s = √(85,000 ÷ 9) = √9,444.4444 = 97.1825
Subtract the mean and divide by the standard deviation
z = (x − x̄) ÷ s
z = (1,310 − 1,160) ÷ 97.1825 = 150 ÷ 97.1825 = 1.5435
Read the area under the normal curve
P(Z < z) from the standard normal table
P(Z < 1.5435) = 0.9386, so the value is at about the 93.86th percentile if the data is roughly normal.
P(Z > 1.5435) = 0.0614. The value is 1.54 standard deviations above the mean.
| # | x | x − x̄ | (x − x̄)² | z |
|---|---|---|---|---|
| 1 | 1,050 | −110 | 12,100 | −1.1319 |
| 2 | 1,180 | 20 | 400 | 0.2058 |
| 3 | 1,220 | 60 | 3,600 | 0.6174 |
| 4 | 990 | −170 | 28,900 | −1.7493 |
| 5 | 1,310 | 150 | 22,500 | 1.5435 |
| 6 | 1,130 | −30 | 900 | −0.3087 |
| 7 | 1,260 | 100 | 10,000 | 1.029 |
| 8 | 1,090 | −70 | 4,900 | −0.7203 |
| 9 | 1,170 | 10 | 100 | 0.1029 |
| 10 | 1,200 | 40 | 1,600 | 0.4116 |
| Σ | 11,600 | 0 | 85,000 |
Rows with |z| above 3 are highlighted. The Σ row of x shows Σx.
What is a z-score?
A z-score (standard score) tells you how many standard deviations a value is above or below the mean. It is calculated as z = (x − μ) ÷ σ. A z-score of 0 is exactly average, +1 is one standard deviation above, and −2 is two below. Because z-scores remove the original units, they let you compare values from different scales, such as an SAT score and an ACT score, or a child’s height and weight. When the data is roughly normal, the z-score also gives a percentile: the share of values below it, read from the standard normal distribution (the z-table). This calculator does both: it works out z for a single value, with left-tail, right-tail and two-tailed probabilities, and it standardises a whole data set using its own mean and standard deviation.
How to calculate a z-score
- In the single value box, enter the raw score x, the mean and the standard deviation. Or paste a data set and press “Use data set mean and SD”.
- Read the z-score, the percentile P(Z < z) and the right-tail and two-tailed probabilities.
- To standardise a whole data set, paste the values in the data box. The table lists the z-score of every value, using the sample or population standard deviation you choose.
- Copy the z-scores as a column to paste back into a spreadsheet.
Z-score formulas
- Z-score (population)
z = (x − μ) ÷ σ
Distance from the mean in standard deviations.
- Z-score (sample)
z = (x − x̄) ÷ s
Uses the sample mean and sample standard deviation.
- Raw score from z
x = μ + z × σ
Turns a z-score back into the original units.
- Percentile
percentile = P(Z < z) × 100
Area under the standard normal curve to the left of z.
Why use this z-score calculator
- Every probability at onceLeft tail, right tail, the area between −z and +z and the two-tailed p-value appear together, so you never have to flip a z-table around.
- A whole data set in one goPaste a column and get the z-score of every value, with outliers flagged, ready to copy back into a spreadsheet.
- Working shown with your numbersEvery answer comes with the formula and each line of arithmetic filled in with your data, plus a table of deviations, so you can check homework or copy the method into an exam answer.
- Paste straight from a spreadsheetCopy a column from Excel, Google Sheets or Numbers and paste it. Commas, spaces, tabs and new lines all work, headers like “Score” are skipped, and multi-column pastes let you pick one column.
- See the spread, not just the numberA histogram with the mean and a one standard deviation band, plus a box plot, shows how the data is spread and where any outliers sit.
- Private, free and instantCalculations run in your browser as you type. Nothing is sent to a server, there is no sign-up, no limit on how many numbers you enter and no “show steps” paywall.
Common uses for z-scores
- Grades and test scores. See how spread out a class’s scores are and how far one score sits from the average.
- Comparing different tests. Put SAT, ACT and class test results on the same scale.
- Spotting outliers. Values with |z| above 3 stand out in any data set.
- Statistics homework. Check answers for AP Statistics, intro stats and lab reports, and copy the step-by-step working.
- Growth charts and health. Read how a child’s height or weight compares with reference values.
- Quality control. Track how much part sizes, fill weights or process times vary, and spot values outside the usual range.
The same statistics in Excel, Google Sheets and a TI-84
To check an answer or redo it in a spreadsheet, use these functions (they work the same in Google Sheets). On a TI-83 or TI-84, enter the data in L1, press STAT, CALC, 1-Var Stats; for a frequency table put the counts in L2 and set FreqList to L2.
| Statistic | Excel and Google Sheets | TI-84 (1-Var Stats) |
|---|---|---|
| Sample standard deviation (s) | =STDEV.S(A2:A20) | Sx |
| Population standard deviation (σ) | =STDEV.P(A2:A20) | σx |
| Sample variance (s²) | =VAR.S(A2:A20) | Sx² |
| Population variance (σ²) | =VAR.P(A2:A20) | σx² |
| Mean | =AVERAGE(A2:A20) | x̄ |
| Median | =MEDIAN(A2:A20) | Med |
| Mode | =MODE.MULT(A2:A20) | Not shown |
| Q1 and Q3 | =QUARTILE.INC(A2:A20, 1) and 3, or QUARTILE.EXC | Q1, Q3 (textbook method) |
| Sum of squares (SS) | =DEVSQ(A2:A20) | Not shown (Σx² is) |
| Mean absolute deviation | =AVEDEV(A2:A20) | Not shown |
| Standard error of the mean | =STDEV.S(A2:A20)/SQRT(COUNT(A2:A20)) | Sx ÷ √n |
| Z-score of a value | =STANDARDIZE(x, mean, sd) | (x − x̄) ÷ Sx |
| Percentile from a z-score | =NORM.S.DIST(z, TRUE) | normalcdf(−1E99, z) |
What this calculator does that most others leave out
We compared the top standard deviation, variance, z-score and quartile calculators in October 2026. Most return one number for one mode, show the formula without your numbers, or cap the number of values. This one was built to cover the gaps.
- Both versions at once. Most calculators make you pick sample or population first. Here s, σ, s² and σ² are always on screen.
- Working with your own numbers. Not just the formula: the mean, every squared deviation, the sum of squares and the final square root are written out with your data.
- Frequency and grouped data. Enter value and count pairs or class intervals instead of typing repeated numbers.
- Quartiles that match your class. Choose the textbook / TI-84 method or Excel’s QUARTILE.INC or QUARTILE.EXC.
- Z-scores and outliers. Every value gets a z-score, and values beyond 1.5 × IQR or 3 standard deviations are flagged.
- No limits, fully private. Paste thousands of values or open a CSV file. Everything is processed on your device and never sent anywhere.
More math and statistics tools
Only need the mean, median and mode, or a weighted average for grades? Use the average calculator. For percentages and percent change, try the percentage calculator, and to type formulas like σ = √(Σ(x − μ)² ÷ N) for a report, use the online math editor. Browse all student tools and calculators.
Worked example: how good is a score of 650?
A test has a mean of 500 and a standard deviation of 100. For a score of 650: z = (650 − 500) ÷ 100 = 1.5. P(Z < 1.5) = 0.9332, so the score is at about the 93rd percentile: better than about 93% of test takers if scores are normally distributed. With the data set preloaded above (10 scores, mean 1160, sample SD 97.1825), a score of 1310 has z = 150 ÷ 97.1825 = 1.5435, about the 94th percentile.
What to expect from the result
Percentiles and probabilities assume the data follows a normal distribution; for skewed data the z-score is still correct but the percentile is only an approximation. Probabilities are computed with a double-precision normal CDF, more precise than a printed z-table. The standard deviation must be greater than 0.
Z-Score Calculator questions
What is the z-score formula?
z = (x − μ) ÷ σ for a population, or z = (x − x̄) ÷ s when you use a sample’s mean and standard deviation. Subtract the mean from the value, then divide by the standard deviation.
How do I convert a z-score to a percentile?
Find the area under the standard normal curve to the left of z, P(Z < z), and multiply by 100. z = 0 is the 50th percentile, z = 1 the 84th, z = 1.645 the 95th and z = 2 about the 97.7th. The calculator does this for you.
What is a good z-score?
It depends on the context. Above 0 is above average. Beyond ±2 is unusual (about 5% of normal data), and beyond ±3 is rare (0.3%) and often treated as an outlier.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. z = −1.2 is 1.2 standard deviations below average, around the 12th percentile.
Should I use the sample or population standard deviation for z-scores?
Use the population σ when the mean and SD describe the whole population, as with published test norms. For z-scores within your own sample, most textbooks and Excel’s STANDARDIZE use whatever SD you supply; choose Sample to use s or Population to use σ.