Suppose we have a set of blood pressures with a mean of 120 Systolic, and a sample standard deviation of 20 points. If we assume a normal distribution of Systolic blood pressures, then between what two values can we be assured 99.7% of all Systolic blood pressures will lie?
99.7% = Mean ± 3 SD
To find the range within which 99.7% of all systolic blood pressures will lie, we can use the concept of the empirical rule (also known as the 68-95-99.7 rule) for a normal distribution.
According to the empirical rule, for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
In this case, the mean systolic blood pressure is 120 and the sample standard deviation is 20.
To find the range for 99.7% of the systolic blood pressures, we need to consider three standard deviations above and below the mean.
The lower end of the range can be found by subtracting three times the standard deviation from the mean:
Lower end = mean - (3 * standard deviation)
Lower end = 120 - (3 * 20)
Lower end = 120 - 60
Lower end = 60
The upper end of the range can be found by adding three times the standard deviation to the mean:
Upper end = mean + (3 * standard deviation)
Upper end = 120 + (3 * 20)
Upper end = 120 + 60
Upper end = 180
Therefore, we can be assured that 99.7% of all systolic blood pressures will lie between 60 and 180.