If I have the mean and standard deviation of data how do I work out the percentage of data within 1,2,and 3 standard deviations of the mean?
:) any help would be much appreciated
thanks
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
Thanks :)
To work out the percentage of data within a certain number of standard deviations from the mean, you can use the empirical rule (also known as the 68-95-99.7 rule). This rule states that for a normal distribution:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
To calculate the percentages, you need the mean and standard deviation of your data set. Here's how to do it:
1. Calculate the lower and upper boundaries for each range.
- For 1 standard deviation, the lower boundary is mean - standard deviation, and the upper boundary is mean + standard deviation.
- For 2 standard deviations, the lower boundary is mean - (2 * standard deviation), and the upper boundary is mean + (2 * standard deviation).
- For 3 standard deviations, the lower boundary is mean - (3 * standard deviation), and the upper boundary is mean + (3 * standard deviation).
2. Calculate the cumulative percentage for each range.
- For 1 standard deviation, subtract the lower boundary from the upper boundary and divide by the total number of data points. Multiply by 100 to get the percentage.
- For 2 standard deviations, subtract the lower boundary from the upper boundary and divide by the total number of data points. Multiply by 100 to get the percentage.
- For 3 standard deviations, subtract the lower boundary from the upper boundary and divide by the total number of data points. Multiply by 100 to get the percentage.
It's important to note that the percentages provided by the empirical rule are approximations and assume that your data follows a normal distribution. In practice, the percentage of data within each range may vary slightly from these approximations.