Using the data given in the table, calculate the standard deviation of the number of children. If necessary, give your answer to three decimal places.
The table shows us there were 15 families with one child, 26 families with two children, three families with three children, 28 families with four children, and 14 families with five children. In total, there were 86 families in the table.
The standard deviation, from a frequency table, can be calculated by square rooting the sum of 𝑓 multiplied by 𝑥 minus 𝜇 all squared divided by the sum of 𝑓. Well, we first calculate 𝜇, the mean of the data.
In our example, if we let the number of children be 𝑥 and the number of families be 𝑓, the frequency, then we can calculate the mean by dividing the sum of 𝑓 multiplied by 𝑥 by the sum of 𝑓. The 𝑓 multiplied by 𝑥 column can be completed by multiplying one by 15, two by 26, three by three, four by 28, and five by 14. This gives us answers of 15, 52, nine, 112, and 70. The sum of these is equal to 258. We can then calculate the mean by dividing 258 by 86. This gives us an answer of three. Therefore, the mean for the data in the table is equal to three.
Our next step is to calculate 𝑥 minus 𝜇, the mean, all squared. The first value will be one minus three all squared. This is equal to four, as negative two squared is four. The second row will be calculated by subtracting the mean from two and then squaring this answer. Two minus three is negative one. And negative one squared is equal to one. We repeat this process to give us values of zero, one, and four, in the third, fourth, and fifth row.
We now need to multiply each of these values by the frequency. In this case, the number of families. 15 multiplied by four is equal to 60. 26 multiplied by one is 26. Three multiplied by zero is zero. 28 multiplied by one is 28. And 14 multiplied by four is 56. The sum of this column is equal to 170.
We can now calculate the standard deviation by square rooting 170 divided by 86. This is equal to 1.406, to three decimal places. The standard deviation of the number of children is 1.406.