math question

This week we will discuss the Empirical Rule.

The empirical rule allows you to determine the proximity of the data to the mean.

This only works for bell shape or symmetric distributions.

• The interval that is one standard deviation away contains approximately

68% of the data.

(�̅� ± 1 *SD)

• The interval that is two standard deviations away contains approximately

95% of the data.

(�̅� ± 2 *SD)

• The interval that is three standard deviations away contains approximately

99.7% of the data.

(�̅� ± 3 *SD)

Let’s continue to look at the Data from Week 2.

Car Price:

Observation 1 $ 20,000 Observation 2 $ 25,000

Observation 3 $ 30,000 Observation 4 $ 31,000

Observation 5 $ 22,500

Observation 6 $ 25,000 Observation 7 $ 29,500

Observation 8 $ 24,000 Observation 9 $ 24,500

Observation 10 $ 25,000

Mean: $ 25,650 Median: $ 25,000

SD: $ 3,488.47 Sample Size: 10

Using the Empirical Rule calculate how many data points fall within the 1, 2 and 3

SD’s?

1) First, we will need to calculate each interval.

25, 650 – 3,488.47 = $22,162 -> round to the nearest dollar 25, 650 + 3,488.47 = $29,138 -> round to the nearest dollar The interval for approximal 68% of the data is ($22,162, $29,138). But how many data points fall within this interval? We see that observations, 1, 2, 5, 6, 8 ,9, and 10. 7 of the 10 observations fall

within this interval. That is 7

10 = 70% of the data falls within 1 SD. This is very

close to 68%.

2) We will calculate the next interval.

25, 650 – (2) 3,488.47 = $18,673 -> round to the nearest dollar 25, 650 + (2) 3,488.47 = $32,627 -> round to the nearest dollar The interval for approximal 95% of the data is ($18,673, $32,627). But how many data points fall within this interval? We see that observations, 1, 2, 3, 4 5, 6, 7 8 ,9, and 10. All 10 of the 10

observations fall within this interval. That is 10

10 = 100% of the data falls within 2

SD’s. This is very close to 95%. Since this is a smaller data set see that all the data points fall within the first 2 observations is not uncommon. We would expect results like this.

3) But it is still a good idea to calculate the last interval.

25, 650 – (3) 3,488.47 = $15,185 -> round to the nearest dollar 25, 650 + (3) 3,488.47 = $36,115 -> round to the nearest dollar The interval for approximal 99.7% of the data is ($15,185, $36,115). But how many data points fall within this interval? Just like with the last interval all the data points fall within this interval and because this is a small data set the results are as expected.

There are no data points that fall outside this range. There doesn’t appear to be

any outliers in this data set. We also see that the mean and median are close

together. There isn’t a big difference between the two values. Because of these

explanations, this data appears to be normal and have a normal distribution. The

data set does not seem to be skewed, in either direction.

We can see how the SD’s line up along the x-axis and it creates the bell-shaped

curve.