Answer:
First, let's compute the mean (average) and the standard deviation of this data set.
The mean (average) is the sum of all numbers divided by the number of items in the set:
mean = (72 + 75 + 74 + 85 + 84 + 72 + 66) / 7 = 528 / 7 = 75.43 (rounded to two decimal places)
Next, we calculate the standard deviation. This is a measure of the amount of variation or dispersion in a set of values.
1. Find the difference between each data point and the mean, square each difference.
2. Find the average of these squared differences.
3. Take the square root of the result.
For our data set:
- The squared differences are: (72-75.43)^2, (75-75.43)^2, (74-75.43)^2, (85-75.43)^2, (84-75.43)^2, (72-75.43)^2, (66-75.43)^2.
- Sum of these squared differences is: 11.76 + 0.1849 + 2.04 + 91.64 + 73.44 + 11.76 + 89.14 = 279.96.
- The average of these squared differences (variance) is 279.96 / 7 = 39.99.
- Standard deviation is the square root of the variance, √39.99 = 6.32 (rounded to two decimal places).
Now we need to find the percentage of data within 1 standard deviation of the mean. The range for 1 standard deviation from the mean is from (mean - standard deviation) to (mean + standard deviation), or from (75.43 - 6.32) to (75.43 + 6.32), which is roughly 69.11 to 81.75.
Counting the data points within this range, we have: 72, 75, 74, 72. There are 4 out of 7 data points within this range.
To find the percentage, we use the formula (number of items within 1 standard deviation / total number of items) * 100%. In this case, it is (4 / 7) * 100% = 57.14%, which rounds to 57% when rounded to the nearest percent. So, about 57% of data points are within 1 standard deviation of the mean.