One Sample Z Test: Formula, Example, Calculator, SPSS, R, Python and Excel Guide

What Is a One Sample Z Test?

A One Sample Z Test is a hypothesis test for one population mean. It compares the observed sample mean with a hypothesized population mean. The test is called a z test because the test statistic follows the standard normal distribution when the population standard deviation is known and the sampling conditions are satisfied.

In this example, the numeric variable is G3, the final grade score. The sample contains 649 students. The observed mean final grade is 11.9060. The benchmark mean is 11.5. The research question is: Is the mean final grade significantly greater than 11.5?

The One Sample Z Test is different from the One Proportion Z Test. A one proportion z test is used for a binary outcome such as yes/no. A one sample z test is used for a numeric mean such as a test score, grade, height, income, time, weight or measurement value.

When Should You Use a One Sample Z Test?

Use a One Sample Z Test when you have one numeric sample, one hypothesized mean and a known population standard deviation. This test is common in quality control, standardized testing, industrial measurement, population benchmark testing and situations where sigma is already established from historical population data.

One Sample Z Test Formula and Hypotheses

The One Sample Z Test formula compares the sample mean with the hypothesized mean using the known population standard deviation:

z = (x-bar - μ0) / (σ / sqrt(n))

Here, x-bar is the observed sample mean, μ0 is the hypothesized population mean, σ is the known population standard deviation and n is the sample size.

Hypotheses for This Worked Example

Conditions for One Sample Z Test

The conditions for One Sample Z Test must be checked before reporting the result. The most important condition is that the population standard deviation must be known. If it is estimated from the sample, a one-sample t test is normally preferred.

Dataset and Variable Used

This tutorial uses the student-por.csv student performance dataset. The main variable is G3, the final grade. The test compares the sample mean of G3 with the hypothesized mean μ0 = 11.5.

External dataset source: UCI Machine Learning Repository: Student Performance dataset.

Verified R, Python and SPSS Results

The One Sample Z Test was reproduced in R, Python and SPSS. The corrected SPSS workflow used the cleaned file spss_ready_data.csv, which imported G3 correctly with 649 valid cases. The SPSS result matches the R and Python outputs.

Main One Sample Z Test Result

Confidence Interval

Descriptive Group Context

R Charts for One Sample Z Test

The R workflow produced visual explanations of the G3 distribution, confidence interval, z statistic, assumptions, numeric-variable context, group means and p-value summary.

1. G3 Distribution with Mean and Hypothesized Mean

The distribution chart shows how final grades are spread across the sample. The observed mean is slightly to the right of the hypothesized mean of 11.5. This difference becomes statistically significant because the sample size is large and the standard error is small.

2. Confidence Interval vs Null Mean

The confidence interval lies above the hypothesized mean. This supports the same conclusion as the z test: the mean final grade is significantly greater than 11.5.

3. Z Statistic Normal Curve

The z statistic is 3.202, which falls in the right tail of the standard normal distribution. The right-tailed p-value is 0.000682, so it is reported as p < .001.

4. One Sample Z Test Condition Check

The condition chart highlights two key points: the sample size is large, and sigma is treated as known. The large sample size supports the normal approximation, while the known sigma condition is essential for calling this a z test.

5. Candidate Numeric Variable Means

This chart shows why G3 is a good teaching variable for a one-sample mean test. It is numeric, meaningful and directly interpretable as a final grade score.

6. Mean G3 by School

The school chart provides descriptive context. GP has a higher mean final grade than MS in this dataset. This is not the main one-sample z test, but it helps readers understand the dataset structure.

7. Mean G3 by Sex

The sex chart shows that female students have a higher mean final grade than male students descriptively. Again, this is contextual and not the main hypothesis test.

8. P-Value Summary

The p-value summary makes the decision clear. The main right-tailed p-value is far below .05, so the null hypothesis is rejected.

9. Formula Summary Panel

The formula panel connects the raw result to the calculation. It shows how the mean difference is divided by the known-sigma standard error to produce the z statistic.

Python Validation Charts for One Sample Z Test

The Python workflow independently validates the same result. It reproduces the observed mean, confidence interval, normal curve, assumptions, group context and formula summary.

1. Python Observed Mean vs Hypothesized Mean

This chart gives the simplest visual explanation of the test. The observed mean is higher than the benchmark mean of 11.5. The z test determines whether this difference is statistically meaningful.

2. Python G3 Distribution with Mean and μ0

The Python distribution chart confirms the same pattern as R. The sample mean is above the null mean, and the large sample size makes the difference precise enough to be significant.

3. Python Confidence Interval vs Null Mean

The confidence interval is approximately [11.6575, 12.1545]. Since the whole interval is above 11.5, it supports the greater-than conclusion.

4. Python Z Statistic Normal Curve

The Python normal curve shows the right-tail area beyond z = 3.202. This right-tail area is the main p-value for the alternative hypothesis μ > 11.5.

5. Python One Sample Z Test Condition Check

The condition chart reinforces the major teaching point: this test is a z test because sigma is treated as known. With an unknown population standard deviation, a one-sample t test should usually be used.

6. Python Candidate Numeric Variable Means

This chart helps readers see that the dataset contains several numeric variables, but G3 is the best main example because it is an interpretable final-grade outcome.

7. Python Mean G3 by School

The Python school chart confirms the same descriptive group pattern as SPSS and R. GP has a higher mean final grade than MS in this dataset.

8. Python Mean G3 by Sex

The Python sex chart confirms that female students have a higher mean G3 than male students descriptively. This chart is included for context, not as the main z test.

9. Python P-Value Summary

The p-value chart confirms that the right-tailed p-value is below .001. The two-sided p-value is also significant at .001364.

10. Python Formula Summary Panel

The formula panel validates every numerical part of the test. It shows that the result is driven by x-bar = 11.9060, μ0 = 11.5, sigma = 3.23, SE = 0.126789 and z = 3.202.

How to Run One Sample Z Test in SPSS, R, Python and Excel

One Sample Z Test in SPSS

SPSS has a built-in one-sample t test, but not a direct menu table for a one-sample z test for a mean. Therefore, the clean SPSS method is to import spss_ready_data.csv, aggregate the sample mean and sample size, enter the known sigma, and manually compute the z statistic and p-value.

* One Sample Z Test in SPSS using cleaned CSV.

GET DATA
 /TYPE=TXT
 /FILE='D:\low kda score priority basis posts\first post\One Sample Z Test\spss_ready_data.csv'
 /ENCODING='UTF8'
 /DELCASE=LINE
 /DELIMITERS=","
 /QUALIFIER='"'
 /ARRANGEMENT=DELIMITED
 /FIRSTCASE=2
 /VARIABLES=
 school A240 sex A240 age F12.4 address A240 famsize A240 Pstatus A240
 Medu F12.4 Fedu F12.4 Mjob A240 Fjob A240 reason A240 guardian A240
 traveltime F12.4 studytime F12.4 failures F12.4 schoolsup A240 famsup A240
 paid A240 activities A240 nursery A240 higher A240 internet A240 romantic A240
 famrel F12.4 freetime F12.4 goout F12.4 Dalc F12.4 Walc F12.4
 health F12.4 absences F12.4 G1 F12.4 G2 F12.4 G3 F12.4.
EXECUTE.

AGGREGATE
 /OUTFILE=* MODE=ADDVARIABLES
 /BREAK=
 /N_G3=N(G3)
 /MEAN_G3=MEAN(G3)
 /SD_G3=SD(G3).

COMPUTE mu0 = 11.5.
COMPUTE known_sigma = 3.23.
COMPUTE se_known = known_sigma / SQRT(N_G3).
COMPUTE z_statistic = (MEAN_G3 - mu0) / se_known.
COMPUTE p_value_right = 1 - CDF.NORMAL(z_statistic, 0, 1).
COMPUTE p_value_two_sided = 2 * (1 - CDF.NORMAL(ABS(z_statistic), 0, 1)).
EXECUTE.

LIST VARIABLES=N_G3 MEAN_G3 SD_G3 mu0 known_sigma se_known z_statistic p_value_right p_value_two_sided
 /CASES=FROM 1 TO 1.

One Sample Z Test in R

In R, the one sample z test can be calculated manually from the mean, known sigma and sample size.

student <- read.csv("student-por.csv")

x <- as.numeric(student$G3)
x <- x[!is.na(x)]

n <- length(x)
xbar <- mean(x)
mu0 <- 11.5
known_sigma <- 3.23

se_known <- known_sigma / sqrt(n)
z_value <- (xbar - mu0) / se_known

p_value_right <- pnorm(z_value, lower.tail = FALSE)
p_value_two_sided <- 2 * pnorm(-abs(z_value))

ci_low <- xbar - qnorm(.975) * se_known
ci_high <- xbar + qnorm(.975) * se_known

n
xbar
se_known
z_value
p_value_right
p_value_two_sided
ci_low
ci_high

One Sample Z Test in Python

Python can reproduce the same test with pandas for data handling and SciPy for the standard normal p-value.

import pandas as pd
import math
from scipy.stats import norm

student = pd.read_csv("student-por.csv")

x = pd.to_numeric(student["G3"], errors="coerce").dropna()

n = len(x)
xbar = x.mean()
mu0 = 11.5
known_sigma = 3.23

se_known = known_sigma / math.sqrt(n)
z_value = (xbar - mu0) / se_known

p_value_right = norm.sf(z_value)
p_value_two_sided = 2 * norm.sf(abs(z_value))

ci_low = xbar - norm.ppf(.975) * se_known
ci_high = xbar + norm.ppf(.975) * se_known

print(n, xbar, se_known, z_value, p_value_right, p_value_two_sided, ci_low, ci_high)

One Sample Z Test in Excel

Excel is a good way to understand the calculator logic behind a one sample z test. Suppose G3 values are in cells A2:A650.

One Sample Z Test Calculator Logic

A One Sample Z Test calculator usually asks for the sample mean, hypothesized mean, known population standard deviation, sample size and alternative hypothesis. For this example, the inputs are:

The calculator should return approximately SE = 0.126789, z = 3.202254, right-tailed p = 0.000682 and 95% z CI = [11.657508, 12.154510].

How to Report One Sample Z Test

A good report should include the test name, variable, sample size, sample mean, hypothesized mean, known sigma, standard error, z statistic, p-value, confidence interval and decision.

Common Mistakes in One Sample Z Test

1. Using a z test when sigma is unknown

The most common mistake is using a one sample z test when the population standard deviation is not known. If sigma is estimated from the sample, a one-sample t test is usually the correct test.

2. Confusing sample SD with known sigma

The sample SD is a descriptive statistic. Known sigma is a population value used in the denominator of the z statistic. In this article, sigma = 3.23 is treated as known for teaching purposes.

3. Using the wrong alternative hypothesis

The p-value depends on the alternative hypothesis. This example uses a right-tailed test because the question asks whether the mean is greater than 11.5.

4. Reporting p = .000

If software displays .000, write p < .001. In this example, the right-tailed p-value is 0.000682, which is correctly reported as p < .001.

5. Treating descriptive group charts as the main test

The school and sex charts are descriptive context only. The main hypothesis test is the overall mean of G3 against μ0 = 11.5.

Related Data Analysis Guides

For related z-test and assumption-checking topics, read the One Proportion Z Test, Kolmogorov-Smirnov Test, Lilliefors Test, DAgostino Pearson Test, Cramer von Mises Test, Brown-Forsythe Test and Hartley F Max Test.

Download SPSS Output and Verification Files

The SPSS output PDF verifies the cleaned-data import, descriptive statistics, reference t test, group means, manual one-sample z test result and formula components.

External References

This guide uses verified R, Python and SPSS outputs together with standard statistical documentation and software references. These resources help readers understand one-sample mean testing, z statistics, p-values and confidence intervals.

• UCI Machine Learning Repository: Student Performance dataset
• IBM SPSS Statistics documentation
• The Comprehensive R Archive Network
• SciPy normal distribution documentation
• Microsoft Excel support documentation