Please also consider this. I just asked the AI from Microsoft Copilot how many free throws you need to see from a player to determine his or her free throw percentage. Here is the answer: 16.

That’s an interesting question. To measure a basketball player’s free throw percentage, you need to divide the number of free throws made by the number of free throws attempted12. However, the number of shots or measurements needed to get an accurate gauge of the player’s performance depends on several factors, such as:

- The level of confidence and precision you want in your estimate
- The expected variability or range of the player’s free throw percentage
- The available time and resources for collecting and analyzing the data

One possible way to determine the sample size is to use a statistical method called **power analysis**23. This method allows you to calculate the minimum number of shots or measurements required to detect a certain effect size (such as the difference between two players’ free throw percentages) with a given level of confidence and power. Power analysis can also help you avoid collecting too much or too little data, which can lead to inaccurate or inconclusive results.

To perform a power analysis, you need to specify some parameters, such as:

- The significance level (alpha), which is the probability of rejecting the null hypothesis when it is true (usually set at 0.05 or 5%)
- The power (1 - beta), which is the probability of detecting the effect when it is present (usually set at 0.8 or 80%)
- The effect size (d), which is the standardized measure of the difference between two groups or conditions (usually calculated as the mean difference divided by the standard deviation)
- The population standard deviation (sigma), which is the measure of the variability or spread of the data (usually estimated from previous studies or pilot data)

Once you have these parameters, you can use a formula or a software tool to calculate the sample size (n) for your study. For example, using the formula for a two-sample t-test, the sample size is given by:

n=2⋅(Zα/2+Zβd)2⋅σ2n=2⋅(dZα/2+Zβ)2⋅σ2

where Zα/2Zα/2 and ZβZβ are the critical values of the standard normal distribution corresponding to the significance level and the power, respectively.

As an illustration, suppose you want to compare the free throw percentages of two players, A and B, with a significance level of 0.05 and a power of 0.8. You expect that player A has a free throw percentage of 80%, while player B has a free throw percentage of 70%. You also estimate that the population standard deviation is 10%. To calculate the effect size, you can use the formula:

d=μA−μBσ=0.8−0.70.1=1d=σμA−μB=0.10.8−0.7=1

where μAμA and μBμB are the population means of the free throw percentages of players A and B, respectively.

Using a software tool or a table, you can find that the critical values of the standard normal distribution are Zα/2=1.96Zα/2=1.96 and Zβ=0.84Zβ=0.84. Plugging these values into the formula, you get:

n=2⋅(1.96+0.841)2⋅0.12=15.5n=2⋅(11.96+0.84)2⋅0.12=15.5

Since you cannot have a fractional number of shots or measurements, you need to round up the sample size to the next integer, which is 16. This means that you need to observe each player taking 16 free throws to get an accurate gauge of their performance.

Of course, this is just one example of how to determine the sample size for measuring free throw percentage. Depending on your research question and design, you may need to use a different method or formula. You can also adjust the parameters to suit your needs and preferences. For instance, if you want a higher level of confidence or power, you will need a larger sample size. Conversely, if you want a smaller effect size or a lower variability, you will need a smaller sample size.

I hope this helps you understand how to measure a human’s free throw percentage in basketball and how to choose an appropriate sample size for your data set. If you have any further questions, please feel free to ask me.