A district mathematics test for all third graders had a normal distribution with a mean of 74 and a standard deviation of 11. What percentage of the third graders tested scored with +1 and -1 standard deviation of the mean?
For any normal distribution, about 68% lie between ± 1 SD.
To find the percentage of third graders who scored within +1 and -1 standard deviation of the mean, we need to use the properties of the normal distribution and the z-score.
1. Calculate the z-score: The z-score measures the number of standard deviations a data point is from the mean, and is calculated using the formula:
z = (x - μ) / σ
where x is the data point, μ is the mean, and σ is the standard deviation.
2. Find the z-score for +1 and -1 standard deviation: Since we want to calculate the percentage within +1 and -1 standard deviation from the mean, we need to find the z-scores for +1 and -1.
For +1 standard deviation:
z1 = (74 + 11) - 74 / 11 = 1
For -1 standard deviation:
z2 = (74 - 11) - 74 / 11 = -1
3. Calculate the area under the standard normal curve: The area under the standard normal curve represents the percentage of data points within a certain range.
Using a standard normal distribution table or a calculator, we can find the area under the curve between z1 and z2.
The area between -1 and 1 standard deviation represents approximately 68% of the data.
Hence, approximately 68% of the third graders tested scored within +1 and -1 standard deviation of the mean.