Test statistics are an important component of statistical hypothesis testing They allow researchers to quantify the evidence against the null hypothesis based on their sample data By calculating test statistics, researchers can determine whether their results are statistically significant and make valid inferences about the population they are studying.
In this comprehensive guide, we will walk through the key steps involved in calculating test statistics, the different types of test statistics, and provide examples to illustrate the calculation process.
What is a Test Statistic?
A test statistic is a number that summarizes the sample data as it relates to the null hypothesis. It provides a standardized measure of how far the sample results diverge from what would be expected under the null hypothesis.
The test statistic incorporates information about the sample mean, sample variance, and sample size. It allows us to determine whether any difference between the sample data and null hypothesis expectations is greater than what would arise from random chance alone.
The basic formula for a test statistic is:
Test Statistic = Signal / NoiseWhere:
- Signal = The difference between the sample statistic and null parameter
- Noise = The variability inherent in the data
If the test statistic turns out to be large, it suggests the signal is strong compared to the background noise. This indicates the null hypothesis is likely false.
On the other hand, a small test statistic suggests a weak signal, meaning any difference between the sample data and null hypothesis could easily be due to chance.
Choosing the Right Test Statistic
There are many different types of test statistics, each designed for specific hypothesis tests. The test statistic you calculate depends on:
- The type of data (continuous, categorical)
- Sample size(s)
- Assumptions made about population distribution
Some common test statistics include:
- t statistic – Used in t-tests and regression analysis
- Z statistic – Used when population variance is known
- F statistic – Used in ANOVA analysis
- Chi-square statistic – Used for tests of categorical data
In practice, statistical software will calculate the appropriate test statistic for you once you specify the type of analysis. But it’s important to understand how these test statistics are derived in order to interpret the output.
Below we explain how to calculate the two most common test statistics – the t statistic and z statistic.
How to Calculate a t Statistic
The t statistic is used when the population variance is unknown and must be estimated from the sample data. This applies to most real-world examples.
The formula for the t statistic depends on whether you are testing:
- One sample mean:
t = (Sample Mean - Null Value) / (Sample Standard Deviation / sqrt(Sample Size))- Difference between two sample means:
t = (Mean 1 - Mean 2) / sqrt(Variance 1/n1 + Variance 2/n2)Let’s walk through an example to illustrate.
A researcher hypothesizes that the average height of 10-year old girls today is greater than the historical average of 56 inches. She takes a random sample of 36 10-year old girls and finds the average height to be 57.5 inches with a sample variance of 25 inches^2.
The null hypothesis is that the population mean height (μ) equals the historical average of 56 inches.
To test this, we first calculate the t statistic:
t = (Sample Mean - Null Value) / (Sample Standard Deviation / sqrt(Sample Size))t = (57.5 - 56) / (5 / sqrt(36)) t = 1.5 / 0.83 = 1.81This t statistic of 1.81 provides evidence against the null hypothesis. We can now use this value to calculate a p-value or compare it to a critical t-value to determine statistical significance.
How to Calculate a Z Statistic
The z statistic is appropriate when the population variance is known. This is rarely the case in practical research, but the z test serves an important role in textbook examples.
The formula for the z statistic is:
z = (Sample Mean - Population Mean) / (Population Standard Deviation / sqrt(Sample Size)Let’s take a look at an example:
A manufacturer produces lightbulbs with an advertised lifespan of 1200 hours. The population standard deviation of lifespans is known to be 150 hours based on extensive quality testing.
A random sample of 64 bulbs from the latest production batch has a mean lifespan of 1180 hours. Management wonders if a production issue has affected the latest batch.
First we state the null hypothesis that the population mean is equal to the advertised 1200 hours.
The z statistic is calculated as:
z = (Sample Mean - Population Mean) / (Population SD / sqrt(Sample Size)) z = (1180 - 1200) / (150 / sqrt(64))z = -1.28This z-score of -1.28 provides evidence against the null hypothesis that the recent production batch has met the advertised 1200 hour average lifespan.
Tips for Calculating Test Statistics
Follow these tips when calculating test statistics:
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Check your data meets the assumptions of the test statistic you intend to use
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Use statistical software like R, Python, SAS, etc. to obtain the test statistic value rather than calculating by hand
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Make sure to define your null hypothesis clearly before choosing and calculating the test statistic
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Report details like the test statistic name, value, degrees of freedom, sample size and p-value when describing your results
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Interpret the test statistic value in context – a larger absolute value indicates more evidence against the null hypothesis
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Do not rely on the test statistic alone – use it to calculate a p-value or compare to a critical value before making conclusions
Common Questions about Test Statistics
What if my data violates assumptions?
Most test statistics rely on certain assumptions being met like normality, equal variances, independence, etc. If your data violates these assumptions, you may need to use alternative tests or data transformations. Non-parametric tests like the Mann-Whitney U or Kruskal-Wallis H are popular options for non-normal data.
Should I calculate test statistics by hand or use software?
It’s strongly recommended to use statistical software like R, SAS, SPSS, etc. to obtain test statistics. This avoids human calculation errors and utilizes optimal computational methods. But you should understand the theory behind each test statistic before applying them through software.
Is a larger test statistic always better?
The larger the absolute value of the test statistic, the stronger the evidence against the null hypothesis. But you can only compare test statistics meaningfully for the same test and same sample sizes. Also be wary of overinflated test statistics due to violations of assumptions or multiple testing.
What happens if I get a negative test statistic value?
Some test statistics like the t and z scores can take on negative values. This simply indicates the sample mean is lower than the value specified in the null hypothesis rather than higher. A negative value still provides evidence against the null, as long as it is a large absolute difference from zero.
Next Steps for Applying Test Statistics
Once you’ve calculated the test statistic, there are two options for interpreting the results:
1. Calculate the p-value
The p-value directly quantifies the statistical significance by representing the probability of obtaining a result at least as extreme as the test statistic under the assumption that the null hypothesis is true.
A smaller p-value indicates stronger evidence to reject the null hypothesis. A p-value below the chosen significance level (e.g. 0.05) means the result is deemed statistically significant.
2. Compare to Critical Values
Each test statistic has an associated critical value table that can be used to establish statistical significance based on the sample size and degrees of freedom.
If the test statistic falls in the critical region outside the specified critical value, the result is considered statistically significant. Comparing to critical values is less common than using p-values but can be helpful for gaining familiarity with how test statistics behave under different conditions.
What is t-statistic and Student’s t-test?
In statistics, the t-statistic, or t-value, is a measure that describes the relationship between a sample and its population. The t-statistic is central to the Students t-test, which is a test for evaluating hypotheses about the population mean.
More precisely, the t-statistic is used to determine whether to support or reject the null hypothesis. It is used in conjunction with the p-value, or critical value, which indicates the probability that your results could have happened by chance. It is comparable to the z-statistic, with the difference being that the t-statistic is applied for small sample sizes or unknown population standard deviations.
How to use this t-statistic calculator?
To compute the t-statistic, you need to provide the following four variables:
- Sample mean, xˉbar xxˉ;
- Population mean, μmuμ;
- Sample size, nnn; and
- Sample standard deviation, sss.
Alternatively, you can use the tool in reverse; for example, you can recover the sample mean from the t-statistic, provided you input all other values.
Lets say you are a basketball player and your game score is 15 (x̄) on average over 36 (n) games, with a standard deviation of 6 (s). You know that an average basketball player scores 10 (μ). Should your performance be considered above average? Or are your scores due to luck? Finding the t-statistic and the probability value will give you some insight. More specifically, finding the t-statistic with the p-value will let you know if there is a significant difference between your mean and the population mean of everyone else.
Applying the previously stated t-statistic formula, you can obtain the following equation.
t=15−106/36=5t = dfrac{15 – 10}{6 / sqrt{36}} = 5t=6/3615−10=5
Now, we know that the t-statistic equals 5, but what does it mean? To gain more knowledge, you should compare this value with a particular threshold (or significance level), lets say 5 percent (α = 5%) of a Student-t distribution. Since the sample size is relatively large (n > 30) we can use the critical value of the standard normal distribution. The critical value of a 5% threshold in a standard normal distribution is 1.645. Since our t-statistic is above the critical value, we can say that you play better than the average.
In fact, we have just performed a Students t-test! Visit our dedicated t-test calculator to learn more.
