Ztest Calculator
This Ztest calculator is a tool that helps you perform a onesample Ztest on the population's mean. Two forms of this test  a twotailed Ztest and a onetailed Ztests  exist, and can be used depending on your needs. You can also choose whether the calculator should determine the pvalue from Ztest, or you'd rather use the critical value approach!
Read on to learn more about Ztest in statistics, and, in particular, when to use Ztests, what is the Ztest formula, and whether to use Ztest vs. ttest. As a bonus, we give some stepbystep examples of how to perform Ztests!
What is a Ztest?
A one sample Ztest is one of the most popular location tests. The null hypothesis is that the population mean value is equal to a given number, μ₀:
H₀: μ = μ₀
We perform a twotailed Ztest if we want to test whether the population mean is not μ₀:
H₁: μ ≠ μ₀,
and a onetailed Ztest if we want to test whether the population mean is less/greater than μ₀:
H₁: μ < μ₀ (lefttailed test); and
H₁: μ > μ₀ (righttailed test).
Let us now discuss the assumptions of a onesample Ztest.
When to use Ztests?
You may use a Ztest if your sample consists of independent data points and:

the data is normally distributed, and you know the population variance;
or

the sample is large, and data follows a distribution which has a finite mean and variance. You don't need to know the population variance.
The reason these two possibilities exist is that we want the test statistics that follow the standard normal distribution N(0,1). In the former case, it is an exact standard normal distribution, while in the latter, it is approximately so, thanks to the central limit theorem.
The question remains, "When is my sample considered large?" Well, there's no universal criterion. In general, the more data points you have, the better the approximation works. Statistics textbooks recommend having no fewer than 50 data points, while 30 is considered the bare minimum.
Ztest formula
Let x_{1}, ..., x_{n}
be an independent sample following the normal distribution N(μ, σ²), i.e., with a mean equal to μ
, and variance equal to σ²
.
We pose the null hypothesis, H₀: μ = μ₀
We define the test statistic, Z, as:
Z = (x̄  μ_{0}) * √n / σ
where:

x̄
is the sample mean, i.e.,x̄ = (x_{1} + ... + x_{n}) / n
; 
μ_{0}
is the mean postulated in H₀; 
n
is sample size; and 
σ
is population standard deviation.
In what follows, the uppercase Z
stands for the test statistic (treated as a random variable), while the lowercase z
will denote an actual value of Z
, computed for a given sample drawn from N(μ,σ²).
If H₀ holds, then the sum S_{n} = x_{1} + ... + x_{n}
follows the normal distribution, with mean n * μ_{0}
and variance n² * σ
. As Z
is the standardization (zscore) of S_{n}/n
, we can conclude that the test statistic Z
follows the standard normal distribution N(0,1), provided that H₀ is true.
If our data does not follow a normal distribution, or if the population standard deviation is unknown (and thus in the formula for Z
we substitute the population standard deviation σ
with sample standard deviation), then the test statistics Z
is not necessarily normal. However, if the sample is sufficiently large, then the central limit theorem guarantees that Z
is approximately N(0,1).
In sections below, we will explain to you how to use the value of the test statistic, z
, to make a decision, whether or not you should reject the null hypothesis. Two approaches can be used in order to arrive at that decision: the pvalue approach, and critical value approach  and we cover both of them! Which one should you use? In the past, the critical value approach was more popular because it was difficult to calculate pvalue from Ztest. However, with help of modern computers, we can do it fairly easily, and with decent precision. In general, you are strongly advised to report the pvalue of your tests!
pvalue from Ztest
Formally, the pvalue is the smallest level of significance at which the null hypothesis could be rejected. More intuitively, pvalue answers the questions:
provided that I live in a world where the null hypothesis holds, how probable is it that the value of the test statistic will be at least as extreme as the z
value I've got for my sample? Hence, a small pvalue means that your result is very improbable under the null hypothesis, and so there is strong evidence against the null hypothesis  the smaller the pvalue, the stronger the evidence.
To find the pvalue, you have to calculate the probability that the test statistic, Z
, is at least as extreme as the value we've actually observed, z
, provided that the null hypothesis is true. (The probability of an event calculated under the assumption that H_{0} is true will be denoted as Pr(event  H_{0})
.) It is the alternative hypothesis which determines what more extreme means:

Twotailed Ztest: extreme values are those whose absolute value exceeds
z
, so those smaller thanz
or greater thanz
. Therefore, we have:pvalue = Pr(Z ≤ z  H_{0}) + Pr(Z ≥ z  H_{0}).
The symmetry of the normal distribution gives:
pvalue = 2 Pr(Z ≤ z  H_{0})

Lefttailed Ztest: extreme values are those smaller than
z
, sopvalue = Pr(Z ≤ z  H_{0})

Righttailed Ztest: extreme values are those greater than
z
, sopvalue = Pr(Z ≥ z  H_{0})
To compute these probabilities, we can use the cumulative distribution function, (cdf) of N(0,1), which for a real number, x
, is defined as:
Also, pvalues can be nicely depicted as the area under the probability density function (pdf) of N(0,1), due to:
Pr(Z ≤ x  H_{0}) = Φ(x) = the area to the left of x
Pr(Z ≥ x  H_{0}) = 1Φ(x) = the area to the right of x
Twotailed Ztest and onetailed Ztest
With all the knowledge you've got from the previous section, you're ready to learn about Ztests.

Twotailed Ztest:
pvalue = Φ(z) + (1  Φ(z))
.From the fact that
Φ(z) = 1  Φ(z)
, we deduce thatpvalue = 2Φ(−z) = 2(1  Φ(z))
;The pvalue is the area under the probability distribution function (pdf) both to the left of
z
, and to the right ofz
: 
Lefttailed Ztest:
pvalue = Φ(z)
;The pvalue is the area under the pdf to the left of our
z
: 
Righttailed Ztest:
pvalue = 1  Φ(z)
;The pvalue is the area under the pdf to the right of
z
:
The decision as to whether or not you should reject the null hypothesis can be now made at any significance level, α, you desire!

if the pvalue is less than, or equal to, α, the null hypothesis is rejected at this significance level; and

if the pvalue is greater than α, then there is not enough evidence to reject the null hypothesis at this significance level.
Ztest critical values & critical regions
The critical value approach involves comparing the value of the test statistic obtained for our sample, z
, to the socalled critical values. These values constitute the boundaries of regions where the test statistic is highly improbable to lie. Those regions are often referred to as the critical regions, or rejection regions. The decision of whether or not you should reject the null hypothesis is then based on whether or not our z
belongs to the critical region.
The critical regions depend on a significance level, &alpha, of the test, and on the alternative hypothesis. The choice of α is arbitrary; in practice, the values of 0.1, 0.05, or 0.01 are most commonly used as α.
Once we agree on the value of α, we can easily determine the critical regions of the Ztest:

Twotailed Ztest:
(∞, Φ^{1}(α/2)] ∪ [Φ^{1}(1  α/2), ∞)

Lefttailed Ztest:
(∞, Φ^{1}(α)]

Righttailed Ztest:
[Φ^{1}(1  α), ∞)
To decide the fate of H₀, check whether or not your z
falls in the critical region:

If yes, then reject H₀ and accept H₁; and

If no, then there is not enough evidence to reject H₀.
As you see, the formulae for the critical values of Ztests involve the inverse, Φ⁻¹, of the cumulative distribution function (cdf) of N(0,1).
How to use the onesample Ztest calculator?
Our calculator reduces all the complicated steps:

Choose the alternative hypothesis: twotailed or left/righttailed.

In our Ztest calculator, you can decide whether to use the pvalue or critical regions approach. In the latter case, set the significance level,
α
. 
Enter the value of the test statistic,
z
. If you don't know it, then you can enter some data that will allow us to calculate yourz
for you: sample mean
x̄
(If you have raw data, go to the average calculator to determine the mean);  tested mean
μ_{0}
;  sample size
n
; and  population standard deviation
σ
(or sample standard deviation if your sample is large).
 sample mean

Results appear immediately below the calculator.
If you want to find z
based on pvalue, please remember that in the case of twotailed tests there are two possible values of z
: one positive and one negative, and they are opposite numbers. This Ztest calculator returns the positive value in such a case. In order to find the other possible value of z
for a given pvalue, just take the number opposite to the value of z
displayed by the calculator.
Ztest examples
To make sure that you've fully understood the essence of Ztest, let's go through some examples:
 A bottle filling machine follows a normal distribution. Its standard deviation, as declared by the manufacturer, is equal to 30 ml. A juice seller claims that the volume poured in each bottle is, on average, one liter, i.e., 1000 ml, but we suspect that in fact the average volume is smaller than that...
Formally, the hypotheses that we set are the following:

H₀: μ = 1000 ml

H₁: μ < 1000 ml
We went to a shop and bought a sample of 9 bottles. After carefully measuring the volume of juice in each bottle, we've obtained the following sample (in milliliters):
1020, 970, 1000, 980, 1010, 930, 950, 980, 980
.

Sample size:
n = 9
; 
Sample mean: `x̄ = 980 ml1;

Population standard deviation:
σ = 30 ml
; 
So
Z = (980  1000) / (30 / √9) = 2
; and 
Therefore,
pvalue = Φ(2) ≈ 0.0455
.
As 0.0455 < 0.05
, we conclude that our suspicions aren't groundless; at the most common significance level, 0.05, we would reject the producer's claim, H₀, and accept the alternative hypothesis, H₁.

We tossed a coin 50 times. We got 20 tails and 30 heads. Is there sufficient evidence to claim that the coin is biased?
Clearly, our data follows Bernoulli distribution, with some success probability
p
and varianceσ² = p * (1p)
. However, the sample is large, so we can safely perform a Ztest. We adopt the convention that getting tails is a success.Let us state the null and alternative hypotheses:

H₀: p = 0.5 (the coin is fair  the probability of tails is
0.5
) 
H₁: p ≠ 0.5 (the coin is biased  the probability of tails differs from
0.5
)

In our sample we have 20 successes (denoted by ones) and 30 failures (denoted by zeros), so:

Sample size
n = 50
; 
Sample mean
x̄ = 20/50 = 0.4
; 
Population standard deviation is given by
σ = √(0.5*0.5)
(because0.5
is the proportionp
hypothesized in H₀). Hence,σ = 0.5
; 
So
Z = (0.4  0.5) / (0.5 / √50) = √2 ≈ 1.4142
; and 
Therefore,
pvalue ≈ 2 * Φ(1.4142) ≈ 0.1573
.
Since 0.1573 > 0.1
we don't have enough evidence to reject the claim that the coin is fair, even at such a large significance level as 0.1
. In that case, you may safely toss it to your Witcher
Ztest vs ttest
Recall that we use a ttest for testing the population mean of a normally distributed dataset which had an unknown population standard deviation. We get this by replacing the population standard deviation in the Ztest statistic formula by the sample standard deviation, which means that this new test statistic follows (provided that H₀ holds) the tStudent distribution with n1 degrees of freedom instead of N(0,1).
For large samples, the tStudent distribution with n degrees of freedom approaches the N(0,1). Hence, as long as there are a sufficient number of data points (at least 30), it does not really matter whether you use the Ztest or the ttest, since the results will be almost identical. However, for small samples with unknown variance, remember to use the ttest instead of Ztest, because in such case the tStudent distribution differs quite significantly from the N(0,1)!