How do I calculate the percentage of data within one standard deviation of the mean?

Question

Using the sample set 4, 9, 10, 13, 16, 17, 18, 22, 23, 38 and knowing that the mean is 17 and the standard deviation for the sample set is 9.439 how do I calculate the percentage of data within one standard deviation of the mean?

Answer 1

First we need to find the Upper Bound & Lower Bound.

Determine the range for one standard deviation:

1. Mean (μ) = 17
2. Standard deviation (σ) = 9.439

The range for one standard deviation from the mean is:

Lower bound=μ−σ=17−9.439≈7.561

Upper bound=μ+σ=17+9.439≈26.439

Identify the data within the bounds we just solved for.

Your dataset is: 4, 9, 10, 13, 16, 17, 18, 22, 23, 38.

The data points within the range of approximately 7.561 to 26.439 are:

9, 10, 13, 16, 17, 18, 22, 23

Count the number of data points within this range:

There are 8 data points within one standard deviation.

1. Calculate the percentage: The total number of data points is 10. (we are only using 8 though)
2. The percentage of data points within one standard deviation is:

Percentage=(Number of points within range/Total number of points)×100

Percentage=(810)×100=80%

So, 80% of the data falls within one standard deviation of the mean.

Answer 2

68.27% of the POPULATION data lies within plus or minus 1 standard deviation from the mean, if that's what you mean to say. If you're asking what % of the data lies between the mean and + 1 standard deviation OR between the mean and - 1 standard deviation, then it would be half of that.

If you are asking about the sample, you would need to go up one standard deviation (so 17 + 9.439 or 26.439) and go down one standard deviation (17 - 9.439 or 7.561) and then inspect the sample data set for outliers. Only "4" and "38" are outside that. Therefore 8/10 of the data values are within plus or minus 1 standard deviation.