Data Matters with Fathom! Dynamic Statistics™ software

Activity 1.3

The law of large numbers claims that as a random sample gets larger, the proportions in the sample tend to approach the probabilities that are generating those proportions. This project lets you check that for yourself.

In Fathom, drag a new collection onto the workspace. Add a new attribute, Trial. Right-click Trial and select New Cases. . . to add about 10 cases. Right-click Trial again, and select Edit Formula. . . . Enter caseIndex in the formula box and click OK. “CaseIndex“ refers to the row of the data table. You are essentially calling the first row Trial 1, the second row Trial 2, and so on.

Add a new attribute, CoinFlip. Set CoinFlip’s formula to randomBinomial(1). This fills CoinFlip with 1’s and 0’s. On each row, the probability of a 1 is 50% and the probability of a 0 is 50%. This is just like flipping a coin. Each 1 indicates a head and each 0 indicates a tail.

Add another attribute, CumulativeSum, with the formula Prev(CumulativeSum) + CoinFlip. Look at the data table. On each row, CumulativeSum has the sum of the CoinFlip values from the first to the current row.

Here’s how CumulativeSum’s formula works: Prev() is the previous value of whatever you type into the parentheses. On each row, Prev(CumulativeSum) is the value of CumulativeSum on the previous row. Fathom adds that row’s value for CoinFlip to the value of CumulativeSum from the previous row.

Add a last attribute, HeadsPercent, with the formula Round(100*CumulativeSum/Trial). On each row, HeadsPercent has the percentage of the flips in that row or above that came up heads (1). Round rounds the percent so you don’t have distracting decimals.

Scan down the column of HeadsPercent. It already shows you the law of large numbers. Although the sample size is small, the proportions range widely, and as the sample size gets larger, the proportions tend to vary closer and closer to 50.

To see a better picture, drag a new graph onto the workspace. Drag Trial to the bottom of the graph and drag HeadsPercent to the y-axis. Select Scatter Plot, then Line Scatter Plot.

To let you more clearly see where 50% is, right-click on the chart and select Plot Function. In the function space, type 50 and click OK. It helps to also plot lines at 40 and 60.

Right-click on the graph and select Show Graph Info. Edit the blue text to set the HeadsPercent axis to go from 0 to 100. Then use “Ctrl-Y” to take a new sample.

You might try adding more cases to see the pattern more clearly.

Report

Print out some of the charts and write a few sentences explaining what this simulation shows you about where to expect proportions to fall when you flip coins.