Reference · Statistics

Z-Score Table

The full standard normal table, plus a lookup that highlights the cell for any z-score.

Look up a z-score

Between −3.49 and 3.49. The matching cell is highlighted below.

Which half of the table

Negative z

Positive z

P(Z ≤ z) — area to the left

0.9750

P(Z > z) — area to the right

0.0250

Two-tailed P(|Z| > z)

0.0500

Standard normal cumulative probabilities for positive z-scores
z	.00	.01	.02	.03	.04	.05	.06	.07	.08	.09
0.0	0.5000	0.5040	0.5080	0.5120	0.5160	0.5199	0.5239	0.5279	0.5319	0.5359
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
0.2	0.5793	0.5832	0.5871	0.5910	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.6480	0.6517
0.4	0.6554	0.6591	0.6628	0.6664	0.6700	0.6736	0.6772	0.6808	0.6844	0.6879
0.5	0.6915	0.6950	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.7190	0.7224
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549
0.7	0.7580	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852
0.8	0.7881	0.7910	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.8340	0.8365	0.8389
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.8770	0.8790	0.8810	0.8830
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.8980	0.8997	0.9015
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319
1.5	0.9332	0.9345	0.9357	0.9370	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9756	0.9761	0.9767
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
2.1	0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9854	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.9890
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916
2.4	0.9918	0.9920	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936
2.5	0.9938	0.9940	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.9960	0.9961	0.9962	0.9963	0.9964
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.9970	0.9971	0.9972	0.9973	0.9974
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.9980	0.9981
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986
3.0	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.9990	0.9990
3.1	0.9990	0.9991	0.9991	0.9991	0.9992	0.9992	0.9992	0.9992	0.9993	0.9993
3.2	0.9993	0.9993	0.9994	0.9994	0.9994	0.9994	0.9994	0.9995	0.9995	0.9995
3.3	0.9995	0.9995	0.9995	0.9996	0.9996	0.9996	0.9996	0.9996	0.9996	0.9997
3.4	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9998

Each cell is the area under the standard normal curve to the left of that z-score. Read the whole-and-tenths digit down the left edge and the hundredths digit across the top: row 1.9 + column .06 gives z = 1.96.

A z-table gives the area under the standard normal curve to the left of a z-score. At z = 1.96 the table reads 0.9750, so 97.50% of values sit below that point and 2.50% above it. Read the tenths down the left edge and the hundredths across the top.

What a z-table actually shows

The standard normal distribution is the bell curve with a mean of 0 and a standard deviation of 1. A z-score says how many standard deviations a value sits from the mean, and the z-table turns that distance into a probability: the proportion of the distribution lying below it. Because the total area under the curve is exactly 1, every entry in the table is between 0 and 1, and the table is the cumulative distribution function evaluated point by point.

The curve is symmetric about 0, which is why the table comes in two halves. The positive half runs from z = 0.00 upward and its entries are all above 0.5; the negative half mirrors it, so the area to the left of −1.96 (0.0250) is exactly the area to the right of +1.96. Once you see that symmetry you only ever need one half, though having both saves a subtraction.

Turning a raw score into a z-score

The table only speaks in z-scores, so a raw measurement has to be standardised first. Subtract the mean and divide by the standard deviation — that strips out the units and puts any normal variable on the same scale the table is built for.

z = (x − μ) ÷ σ

x is the raw value, μ the population mean, σ the population standard deviation

1
Standardise the value. A test score of 128 in a population with μ = 100 and σ = 15 gives z = (128 − 100) ÷ 15 = 1.8667, which rounds to 1.87 for a table lookup.
2
Split the z-score at the decimal point. For z = 1.87 the row is 1.8 (the units and tenths) and the column is .07 (the hundredths).
3
Read the cell where that row and column meet. Row 1.8 and column .07 give 0.9693 — the area to the left of z = 1.87.
4
Convert the area to the probability you need. P(below) = 0.9693 directly; P(above) = 1 − 0.9693 = 0.0307; P(between two z-scores) is the larger area minus the smaller.
5
Interpret it in context. A score of 128 beats about 96.93% of the population, leaving roughly 3.07% above it.

Z-scores worth memorising

Areas to four decimal places. The last two rows are the values behind the 95% and 99% confidence intervals.

z	Area to the left	Area to the right	Two-tailed area
1.00	0.8413	0.1587	0.3173
1.28	0.8997	0.1003	0.2005
1.645	0.9500	0.0500	0.1000
1.96	0.9750	0.0250	0.0500
2.33	0.9901	0.0099	0.0198
2.576	0.9950	0.0050	0.0100

Where students go wrong

The most common mistake is reading the table as the area to the right. Standard z-tables — including this one — are cumulative from the left, so a positive z always returns a number above 0.5. If your answer for a large positive z comes out small, you have taken the complement by accident.

The second is forgetting to double for a two-tailed test. The area beyond z = 1.96 in one tail is 0.0250, but a two-tailed test at the 5% level puts 0.0250 in each tail, which is where the familiar 1.96 comes from. The third is using a z-table at all when the population standard deviation is unknown and the sample is small — that is what the t-distribution exists for.

How do I read a z-table?

Split the z-score at the hundredths place: the units and tenths pick the row, the hundredths pick the column. The cell where they meet is the area under the curve to the left of that z-score. For z = 1.87, row 1.8 and column .07 give 0.9693.

What is the difference between a positive and a negative z-table?

They are two halves of the same distribution. A negative z-table covers scores below the mean and returns areas under 0.5, while the positive half returns areas above 0.5. Because the curve is symmetric, the area left of −z always equals the area right of +z.

Why is 1.96 the number everyone remembers?

The area beyond z = 1.96 is 0.0250 in each tail, which adds to 0.05 across both tails. That makes ±1.96 the cut-off for a two-tailed test at the 5% level and the multiplier in a 95% confidence interval.

How do I find the area between two z-scores?

Look up both and subtract the smaller area from the larger. For z = −1.00 and z = 1.00 that is 0.8413 − 0.1587 = 0.6826, the familiar 68% inside one standard deviation of the mean.

When should I use a t-table instead?

Use the t-distribution when the population standard deviation is unknown and you are estimating it from a small sample. The t-table has an extra degrees-of-freedom dimension and gives wider critical values; as the sample grows the two converge.

What does a z-score above 3 mean?

It is more than three standard deviations from the mean, which happens for roughly 0.13% of a normal distribution in one tail. Standard tables stop near 3.49 because beyond that the areas round to 0.9998 and higher and stop being useful to read off.

Z-Score Table

What a z-table actually shows

Turning a raw score into a z-score

Z-scores worth memorising

Where students go wrong

Related tools

Z-Score Calculator

Normal Distribution

T Distribution Table

Critical Value Calculator