Calculating Standard Deviation Solutions

• The measures of spread show the    variability    contained in the data.

• The measures of spread are:

  •    Range   : The difference between the maximum and minimum value in the data set

  •    Standard deviation   : The measure of how data deviates from the mean

• Standard deviation (𝜎):

• • is a measure of spread based on    the mean of a data set   .

  • of a population represents the average deviation, or    distance of elements from    the population mean.

  • is    always positive    because it is the distance from the mean.

  • is rounded to one decimal place further than the    raw data set   .

Note

The formula for standard deviation of a population is slightly different from the formula for standard deviation, which is not used in this level.

• The standard deviation of a population can be calculated using the formula:
  $\sigma =\sqrt{\frac{\sum _{}{\left(x_i–\mu \right)}^2}{N}}$

To find the standard deviation of a population:

1. Calculate the    mean   , 𝜇.

2. Calculate the    square of the difference    between each data point, $x_i$, and the    mean   : ${\left(x_i–\mu \right)}^2$

3. Find the    sum    of all values in step 2.

4. Calculate the    variance   , ${𝜎}^2$, by dividing the sum by the number of numbers in the data set N.

5. Take the    square root    of the variance.

Example 3

Piano teacher, Ms. Mbuy, tracked the number of students who registered for lessons over ten semesters. She wants to know how the class sizes vary from term to term. The class sizes are:$\left\{5, 6, 6, 7, 7, 8, 8, 9, 9, 10\right\}$.

1. Calculate the standard deviation without technology.
2. Over the same semesters, Mr. Jacono, the strings teacher, determined that his classes have a $\mu =10.9, \sigma =2.1.$ Which teacher’s classes have greater variance? Explain.