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📊Z-Score Calculator

Calculate z-scores, probabilities, and percentiles instantly. Find how many standard deviations a value is from the mean. Free online z-score calculator with z-table, step-by-step solutions, and real-world examples.

Use this calculator to compute the z-score of a normal distribution.

Raw Score, x

Population Mean, μ

Standard Deviation, σ

Z-Table (Area between 0 and Z)

A z-table consists of standardized values used to determine the probability that a given statistic is below, above, or between values in the standard normal distribution. A z-score of 0 is at the center of the curve.

z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.0	0.00000	0.00399	0.00798	0.01197	0.01595	0.01994	0.02392	0.02790	0.03188	0.03586
0.1	0.03983	0.04380	0.04776	0.05172	0.05567	0.05962	0.06356	0.06749	0.07142	0.07535
0.2	0.07926	0.08317	0.08706	0.09095	0.09483	0.09871	0.10257	0.10642	0.11026	0.11409
0.3	0.11791	0.12172	0.12552	0.12930	0.13307	0.13683	0.14058	0.14431	0.14803	0.15173
0.4	0.15542	0.15910	0.16276	0.16640	0.17003	0.17364	0.17724	0.18082	0.18439	0.18793
0.5	0.19146	0.19497	0.19847	0.20194	0.20540	0.20884	0.21226	0.21566	0.21904	0.22240
0.6	0.22575	0.22907	0.23237	0.23565	0.23891	0.24215	0.24537	0.24857	0.25175	0.25490
0.7	0.25804	0.26115	0.26424	0.26730	0.27035	0.27337	0.27637	0.27935	0.28230	0.28524
0.8	0.28814	0.29103	0.29389	0.29673	0.29955	0.30234	0.30511	0.30785	0.31057	0.31327
0.9	0.31594	0.31859	0.32121	0.32381	0.32639	0.32894	0.33147	0.33398	0.33646	0.33891
1.0	0.34134	0.34375	0.34614	0.34849	0.35083	0.35314	0.35543	0.35769	0.35993	0.36214
1.1	0.36433	0.36650	0.36864	0.37076	0.37286	0.37493	0.37698	0.37900	0.38100	0.38298
1.2	0.38493	0.38686	0.38877	0.39065	0.39251	0.39435	0.39617	0.39796	0.39973	0.40147
1.3	0.40320	0.40490	0.40658	0.40824	0.40988	0.41149	0.41308	0.41466	0.41621	0.41774
1.4	0.41924	0.42073	0.42220	0.42364	0.42507	0.42647	0.42785	0.42922	0.43056	0.43189
1.5	0.43319	0.43448	0.43574	0.43699	0.43822	0.43943	0.44062	0.44179	0.44295	0.44408
1.6	0.44520	0.44630	0.44738	0.44845	0.44950	0.45053	0.45154	0.45254	0.45352	0.45449
1.7	0.45543	0.45637	0.45728	0.45818	0.45907	0.45994	0.46080	0.46164	0.46246	0.46327
1.8	0.46407	0.46485	0.46562	0.46638	0.46712	0.46784	0.46856	0.46926	0.46995	0.47062
1.9	0.47128	0.47193	0.47257	0.47320	0.47381	0.47441	0.47500	0.47558	0.47615	0.47670
2.0	0.47725	0.47778	0.47831	0.47882	0.47932	0.47982	0.48030	0.48077	0.48124	0.48169
2.1	0.48214	0.48257	0.48300	0.48341	0.48382	0.48422	0.48461	0.48500	0.48537	0.48574
2.2	0.48610	0.48645	0.48679	0.48713	0.48745	0.48778	0.48809	0.48840	0.48870	0.48899
2.3	0.48928	0.48956	0.48983	0.49010	0.49036	0.49061	0.49086	0.49111	0.49134	0.49158
2.4	0.49180	0.49202	0.49224	0.49245	0.49266	0.49286	0.49305	0.49324	0.49343	0.49361
2.5	0.49379	0.49396	0.49413	0.49430	0.49446	0.49461	0.49477	0.49492	0.49506	0.49520
2.6	0.49534	0.49547	0.49560	0.49573	0.49585	0.49598	0.49609	0.49621	0.49632	0.49643
2.7	0.49653	0.49664	0.49674	0.49683	0.49693	0.49702	0.49711	0.49720	0.49728	0.49736
2.8	0.49744	0.49752	0.49760	0.49767	0.49774	0.49781	0.49788	0.49795	0.49801	0.49807
2.9	0.49813	0.49819	0.49825	0.49831	0.49836	0.49841	0.49846	0.49851	0.49856	0.49861
3.0	0.49865	0.49869	0.49874	0.49878	0.49882	0.49886	0.49889	0.49893	0.49896	0.49900

How to read: For z = 1.12, find row 1.1 and column 0.02 → 0.36864. This means 36.864% of data lies between 0 and 1.12.

What's a Z-score Anyway?

The z-score, also referred to as standard score, z-value, and normal score, is a dimensionless quantity that indicates the signed, fractional number of standard deviations by which an event is above the mean value being measured. Values above the mean have positive z-scores, while values below the mean have negative z-scores.

A z-score tells you how many standard deviations away from the mean your data point is.

Sounds fancy? It's not. Think of it like this:

Imagine you're in a class where the average test score is 75. You got an 85. That's 10 points above average. But is that a big deal? Depends on how spread out the scores are.

If everyone scored between 70 and 80, then yeah - 85 is amazing. But if scores range from 40 to 100, then 85 is just... okay.

The z-score captures this. It tells you not just how far you are from average, but how unusual that distance is.

The Z-score Formula (It's Simpler Than You Think)

Here's the formula in plain English:

Z-score = (Your number - Average) ÷ Standard deviation

In math speak: z = (x - μ) / σ

Where:

x = your data point (the number you're checking)
μ (mu) = the average of all the numbers
σ (sigma) = the standard deviation (how spread out the numbers are)

Let's use our test score example: Your score: 85, Class average: 75, Standard deviation: 5. Z-score = (85 - 75) / 5 = 10 / 5 = 2.0. That means you're 2 standard deviations above the average. Pretty good!

What Does Your Z-score Actually Mean?

Here's where it gets interesting. The z-score tells you a story about your data.

Z-score = 0: You're exactly average. Nothing special, nothing bad. You're the middle of the pack.

Z-score between -1 and 1: About 68% of all data falls here. You're totally normal.

Z-score between -2 and 2: About 95% of data lives here. Still pretty normal.

Z-score between -3 and 3: About 99.7% of data is here. You're getting unusual.

Z-score above 3 or below -3: Less than 0.3% of data lives here. You're a unicorn.

Negative Z-scores: What's the Deal?

This confuses a lot of people. A negative z-score just means you're below average. That's it.

It doesn't mean you're bad or wrong. It just means your number is smaller than the average.

Think about height. If the average height is 5'9" and you're 5'5", your z-score would be negative. You're shorter than average. Nothing wrong with that.

Same with test scores. A negative z-score means you scored below the class average. But that's just information - it tells you where you stand.

Real-World Examples of Z-scores

Example 1: Test Scores

Sarah got a 92 on her math test. Class average: 80, SD: 8. Z-score = (92-80)/8 = 12/8 = 1.5. Sarah is 1.5 SD above average — about the 93rd percentile.

Example 2: Baby Weights

A newborn weighs 7.5 pounds. Average: 7.5, SD: 1.2. Z-score = (7.5-7.5)/1.2 = 0/1.2 = 0. This baby is perfectly average.

Example 3: Company Salaries

Employee earns $95,000. Average: $65,000, SD: $15,000. Z-score = (95K-65K)/15K = 30K/15K = 2.0. Top 2.5% of earners.

Z-scores and Percentiles: The Connection

Here's a cool trick. Once you have a z-score, you can figure out your percentile.

Z=0
50th percentile

Z=1
~84th percentile

Z=2
~98th percentile

Z=3
~99.9th percentile

Z=-1
~16th percentile

Z=-2
~2nd percentile

Our calculator actually shows you this automatically. So you don't need to memorize anything.

Common Mistakes People Make

Mistake #1: Using the Wrong Average. Make sure you're using the population average, not the sample average.

Mistake #2: Forgetting the Standard Deviation. You can't calculate a z-score without it.

Mistake #3: Thinking a Negative Z-score is Bad. It's just below average. Context matters.

Mistake #4: Using Z-scores for Non-Normal Data. If your data isn't bell-shaped, z-scores don't tell you much.

Z-scores vs T-scores: What's the Difference?

Z-scores are for when you know the population standard deviation. Like if you're comparing to all adults in the US.

T-scores are for when you only have a sample and don't know the population standard deviation. Like if you surveyed 100 people from your city.

T-scores are basically z-scores for smaller samples. They're a bit more conservative.

How to Use Our Z-score Calculator

Enter your data point (the number you're checking)
Enter the mean (average) of your data set
Enter the standard deviation
Click calculate

You'll get your z-score instantly. Plus, we show you the probability and percentile. No tables to look up. No formulas to memorize.

Pro Tips for Using Z-scores

Tip 1: Z-scores are great for finding outliers. Above 3 or below -3 = probably an outlier.

Tip 2: Compare z-scores across different data sets — like math vs history test performance.

Tip 3: In Excel: =STANDARDIZE(x, mean, standard_dev).

Tip 4: For 95% confidence interval, z-score = 1.96.

A Fun Fact About Z-scores

The z-score was developed by Karl Pearson in the 1890s. He was trying to figure out how to compare different measurements - like comparing heights to weights. Before z-scores, people couldn't easily compare things measured in different units.

So next time you use a z-score, you're using a tool that's over 130 years old. And it's still one of the most useful stats tools out there.

Key Takeaways

Z = 0 means equal to mean

A z-score of 0 indicates the raw score is identical to the population mean.

68-95-99.7 Rule

68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ of the mean.

Compare Different Scales

Z-scores standardize different measurements, allowing comparison across datasets.

P-values from Z

Use z-tables or the calculator to find probabilities (p-values) for hypothesis testing.

Frequently Asked Questions

1. What does a z-score of 2.5 mean?

A z-score of 2.5 means your data point is 2.5 standard deviations above the average. That's pretty high - only about 0.6% of data falls above this point. You're in the top 1% of whatever you're measuring.

2. Can a z-score be greater than 3?

Yes, absolutely. A z-score can be any number. But z-scores above 3 or below -3 are very rare in a normal distribution. Less than 0.3% of data falls there. If you get a z-score of 4 or 5, you've found something extremely unusual.

3. How do I calculate z-score in Excel?

Use the STANDARDIZE function. The formula is =STANDARDIZE(x, mean, standard_dev). For example, =STANDARDIZE(85, 75, 5) gives you 2. You can also calculate it manually with =(x - mean)/standard_dev.

4. What's the difference between z-score and standard deviation?

Standard deviation measures how spread out your entire data set is. A z-score measures where one specific data point sits within that spread. Think of standard deviation as the ruler, and z-score as where your point lands on that ruler.

5. Is a negative z-score bad?

Not at all. A negative z-score just means you're below the average. Whether that's "bad" depends entirely on what you're measuring. Being below average in blood pressure is good. Being below average in test scores might not be.

6. How do I find probability from a z-score?

You can use a z-score table or our calculator. The probability tells you the chance of randomly picking a value less than your data point. For example, a z-score of 1 has about an 84% probability - meaning 84% of values are below it.

7. What's the z-score for a 95% confidence interval?

The z-score for a 95% confidence interval is 1.96. For 90% it's 1.645, and for 99% it's 2.576. These are the critical values you use when calculating confidence intervals.

8. Can I use z-scores for non-normal data?

Technically yes, but it's not very useful. Z-scores still tell you how far from the mean you are, but the percentile interpretations won't be accurate. Stick to normal distributions for meaningful z-score analysis.

9. How do I interpret a z-score of 0?

A z-score of 0 means your data point is exactly at the average. Nothing special - you're right in the middle. About 50% of data falls below you and 50% above you.

10. What's the z-score for an outlier?

There's no hard rule, but most statisticians consider anything with a z-score above 3 or below -3 as a potential outlier. Some use 2.5 as the cutoff. It depends on how strict you want to be.

11. How do I find the z-score from a percentile?

You can use a z-score table in reverse, or use our calculator's inverse function. For example, the 95th percentile corresponds to a z-score of about 1.645. The 99th percentile is about 2.326.

12. Why does my z-score calculator give a different answer than the table?

This usually happens because of rounding. Calculators use more decimal places than tables. Also, some tables use different conventions (like one-tailed vs two-tailed probabilities). Our calculator uses the most common standard.

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