2Given the data 21, 13, 13, 37, 13, 23, 25, 15:
a. What is the outlier in the data?
b. What is the mean with the outlier?
c. What is the mean without the outlier?
13; 21; 17.6
37; 20; 17.6
37; 17.6; 20
13; 17.6; 21
a. The outlier in the data is 37.
b. The mean with the outlier is (21 + 13 + 13 + 37 + 13 + 23 + 25 + 15)/8 = 20.125.
c. The mean without the outlier is (21 + 13 + 13 + 13 + 23 + 25 + 15)/7 = 17.6.
To find the outlier in the given data, we can use the "1.5 * IQR" rule. The IQR (interquartile range) is calculated by finding the difference between the third quartile (Q3) and the first quartile (Q1). Once we have the IQR, we can determine the lower and upper fences by subtracting 1.5 times the IQR from Q1 and adding 1.5 times the IQR to Q3, respectively. Any value below the lower fence or above the upper fence is considered an outlier.
a. To find the outlier in the data, let's first calculate the quartiles:
Step 1: Sort the data in ascending order: 13, 13, 13, 15, 21, 23, 25, 37.
Step 2: Calculate Q1 (first quartile): (n+1)/4 = (8+1)/4 = 2.25. Since this is not a whole number, interpolate between the 2nd and 3rd values: Q1 = (13+13)/2 = 13.
Step 3: Calculate Q3 (third quartile): 3*(n+1)/4 = 3*(8+1)/4 = 6.75. Again, interpolate between the 6th and 7th values: Q3 = (23+25)/2 = 24.
Step 4: Calculate IQR = Q3 - Q1 = 24 - 13 = 11.
Step 5: Calculate the lower fence: Q1 - 1.5 * IQR = 13 - 1.5 * 11 = -4.5.
Step 6: Calculate the upper fence: Q3 + 1.5 * IQR = 24 + 1.5 * 11 = 41.5.
Outliers are any values below -4.5 or above 41.5. In this data, we have only one value that is above the upper fence, which is 37. Therefore, the outlier in the data is 37.
b. To calculate the mean with the outlier, we sum up all the values and divide by the total count:
Mean = (13 + 13 + 13 + 37 + 13 + 23 + 25 + 15) / 8 = 152 / 8 = 19.
c. To calculate the mean without the outlier, we ignore the outlier value (37) and sum up the remaining values, then divide by the number of remaining values:
Mean = (13 + 13 + 13 + 13 + 23 + 25 + 15) / 7 = 115 / 7 = 16.43 (rounded to two decimal places).
So, the answers are:
a. The outlier in the data is 37.
b. The mean with the outlier is 19.
c. The mean without the outlier is 16.43.
To find the outlier in the given data, we need to identify the value that is significantly different from the rest. One common way to determine an outlier is by using the boxplot method. Here's how you can find the outlier in the data:
1. Arrange the data in ascending order: 13, 13, 13, 15, 21, 23, 25, 37.
2. Calculate the median by finding the middle value of the data. In this case, the median is 15.5, which is the average of the two middle values (15 and 21).
3. Find the lower quartile (Q1) and the upper quartile (Q3). The lower quartile represents the median of the lower half of the data, while the upper quartile represents the median of the upper half of the data.
- Q1 = 13 (the lower median of the lower half of the data)
- Q3 = 23 (the median of the upper half of the data)
4. Calculate the interquartile range (IQR) by subtracting Q1 from Q3:
IQR = Q3 - Q1 = 23 - 13 = 10.
5. Determine the lower and upper fences to determine potential outliers. The lower fence is calculated by subtracting 1.5 times the IQR from Q1, while the upper fence is calculated by adding 1.5 times the IQR to Q3:
- Lower fence = Q1 - 1.5 * IQR = 13 - 1.5 * 10 = -2
- Upper fence = Q3 + 1.5 * IQR = 23 + 1.5 * 10 = 38
6. Identify any values outside the lower and upper fences as potential outliers. In this case, the value of 37 is greater than the upper fence, so it is considered an outlier.
So, the outlier in the given data is 37.
Next, to find the mean with and without the outlier:
To calculate the mean, add up all the values and divide by the total number of values.
a. Mean with the outlier:
Sum of values = 21 + 13 + 13 + 37 + 13 + 23 + 25 + 15 = 160
Total number of values = 8
Mean = 160 / 8 = 20
b. Mean without the outlier:
Sum of values (excluding outlier) = 21 + 13 + 13 + 13 + 23 + 25 + 15 = 123
Total number of values (excluding outlier) = 7
Mean = 123 / 7 = 17.6
c. To summarize:
- The outlier in the data is 37.
- The mean with the outlier is 20.
- The mean without the outlier is 17.6.