Assume that a set of test scores is normally distributed with a mean of 100 and a standard deviation of 20. What percentage of scores is less than 140?

In a normal distribution, approximately 68% of scores are within 1 standard deviation of the mean; half of these (34%) are above the mean, and the other half are below the mean. Approximately 95% (more accurately, 95.45%) of results are within 2 standard deviations of the mean, including those 68% that are within 1 standard deviation.

Since our standard deviation is 20, and we are asked what percentage are below 140 (which is 2 standard deviations above the mean of 120), we are essentially being asked to find what percentage of scores are not above 140, which would equal 100% - (percent of scores above 140).

If 95.45% of scores are within 2 SD of the mean, then approx 4.55% are outside of that range. Of these, half will be more than two standard deviations below the mean (i.e. less than 60), and half will be more than 2 standard deviations above the mean (i.e. greater than 140). Thus to find the percent of scores greater than 140, we will divide 4.55 by 2, coming up with approx 2.725%.

Then, the percent of results below 100% will simply be 100%-2.725% = 97.275%.

Of note, many worksheets, teachers, and textbooks will allow you to simply round that 95.45% I mentioned earlier to 95%, following the 68/95/99.7 rule (e.g. 68% of the data is within 1 SD, 95% within 2 SD, and 99.7% within 3 SD of the mean). If you are instructed to do this, then in the problem above we would say 5% lies more than 2 SD away from the mean, and half of those (2.5%) lie more than 2 SD above the mean. Thus the percentage below 140 in this case would be 100%-2.5% = 97.5%.

Given -

Mean #barx=100#

#sigma=20#

#z=(x-barx)/sigma=(140-100)/20=40/20=2#

What percentage of Score is less than 140?

Probability that a score is less than 140 = Area between#(z=0) # and #(z=oo)# + Area between #(z=0) # and #(z=2)#

Probability that a score is less than 140 # =0.5 +0.4772=0.9772#

Percentage of score less than 140 #= 0.9772 xx 100= 97.72%#