The mean of a set of normally distributed data is 550 and the standard deviation is 35.
What percent of the data is between 515 and 585?
Normalize 515 and 585, namely
z(515)=(515-550)/35=-1
z(585)=(585-550)/35=1
The percent is the difference of probabilities of one-tail z-values between -1 and +1. It should be a little less than 70%.
Well, let me crunch the numbers and juggle some statistics for you! According to my calculations, approximately 68% of the data falls within one standard deviation of the mean in a normal distribution. Since the mean is 550 and the standard deviation is 35, the range of one standard deviation below the mean to one standard deviation above the mean is from 515 to 585. So, it turns out that approximately 68% of the data is between 515 and 585. Remember, statistics can be quite amusing, don't you think?
To find the percentage of the data that falls between 515 and 585 in a normally distributed set, we can use the empirical rule.
According to the empirical rule, in 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, we can calculate the z-scores for both 515 and 585, and then use these z-scores to determine the percentage of data that falls within this range.
First, calculate the z-score for 515:
z = (x - μ) / σ
where x is the value (515), μ is the mean (550), and σ is the standard deviation (35).
z = (515 - 550) / 35
z = -35 / 35
z = -1
Next, calculate the z-score for 585:
z = (x - μ) / σ
where x is the value (585), μ is the mean (550), and σ is the standard deviation (35).
z = (585 - 550) / 35
z = 35 / 35
z = 1
According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean. Therefore, we can determine that the percentage of data between 515 and 585 is the same as the percentage of data within one standard deviation of the mean, which is approximately 68%. Thus, approximately 68% of the data is between 515 and 585.
To find the percentage of data between two values, we need to determine the Z-scores corresponding to those values and then use the Z-table to find the corresponding percentage.
The Z-score can be calculated using the formula:
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the value in question
μ is the mean of the distribution
σ is the standard deviation of the distribution
For the lower value, X = 515:
Z1 = (515 - 550) / 35 = -1
For the upper value, X = 585:
Z2 = (585 - 550) / 35 = 1
Now we need to find the corresponding area under the normal distribution curve between these two Z-scores.
Using a Z-table or a statistical calculator, find the area corresponding to Z1 and Z2.
The Z-table provides the area to the left of a given Z-score. Since we want the area between Z1 and Z2, we need to subtract the area to the left of Z1 from the area to the left of Z2.
From the Z-table, the area to the left of Z1 (-1) is approximately 0.1587, and the area to the left of Z2 (1) is approximately 0.8413.
To find the area between Z1 and Z2, subtract the area to the left of Z1 from the area to the left of Z2:
Area = 0.8413 - 0.1587 = 0.6826
Multiply the result by 100 to get the percentage:
Percentage = 0.6826 * 100 ≈ 68.26%
Approximately 68.26% of the data falls between 515 and 585 in this normally distributed data set.