Given the data 14, 26, 23, 19, 24, 46, 15, 21:
a.
What is the outlier in the data?
b.
What is the mean with the outlier?
c.
What is the mean without the outlier?
a. The outlier in the data is 46.
b. The mean with the outlier is (14+26+23+19+24+46+15+21)/8 = 24.5.
c. The mean without the outlier is (14+26+23+19+24+15+21)/7 = 20.86 (rounded to two decimal places).
Given the data 14, 26, 23, 19, 24, 46, 15, 21:
a. What is the outlier in the data?
b. What is the mean with the outlier?
c. What is the mean without the outlier?
1.14; 20.3; 23.5
2.14; 23.5; 20.3
3.46; 20.3; 23.5
4.46; 23.5; 20.3
3.46; 20.3; 23.5
Given the data 14, 26, 23, 19, 24, 46, 15, 21:
a. What is the outlier in the data?
b. What is the mean with the outlier?
c. What is the mean without the outlier?
14; 20.3; 23.5
14; 23.5; 20.3
46; 20.3; 23.5
46; 23.5; 20.3
3. 46; 20.3; 23.5
a. To find the outlier in the data, we can use the interquartile range (IQR) method. First, we need to find the lower quartile (Q1) and the upper quartile (Q3).
Step 1: Arrange the data in ascending order: 14, 15, 19, 21, 23, 24, 26, 46.
Step 2: Find the median ( Q2), which is the middle value of the data set. In this case, the median is 23.
Step 3: Find the lower quartile (Q1), which is the median of the lower half of the data set. In this case, Q1 is the median of 14, 15, 19, and 21. Thus, Q1 is 17.
Step 4: Find the upper quartile (Q3), which is the median of the upper half of the data set. In this case, Q3 is the median of 24, 26, and 46. Thus, Q3 is 26.
Step 5: Calculate the IQR by subtracting Q1 from Q3. In this case, IQR = Q3 - Q1 = 26 - 17 = 9.
Step 6: Define the outlier using the formula: Q1 - 1.5 * IQR (lower limit) or Q3 + 1.5 * IQR (upper limit). Any number greater than the upper limit or less than the lower limit is an outlier. In this case, the lower limit is 17 - 1.5 * 9 = 3.5, and the upper limit is 26 + 1.5 * 9 = 40.5.
Therefore, the outlier in the data is 46.
b. To find the mean with the outlier, we add up all the data points and divide by the total number of data points.
Mean = (14 + 26 + 23 + 19 + 24 + 46 + 15 + 21) / 8 = 188 / 8 = 23.5.
Therefore, the mean with the outlier is 23.5.
c. To find the mean without the outlier, remove the outlier from the data set and then calculate the mean.
Mean = (14 + 26 + 23 + 19 + 24 + 15 + 21) / 7 = 142 / 7 = 20.29 (rounded to two decimal places).
Therefore, the mean without the outlier is 20.29.
To find the outlier in a set of data, you can use various methods such as the Tukey's fences, the Z-score, or simply by visually inspecting the data for any extreme values. Let's use the Z-score method to find the outlier in the given data.
a. Finding the outlier using Z-score:
Step 1: Calculate the mean of the data set.
To find the mean, add up all the numbers in the data set and divide by the total count.
Mean = (14 + 26 + 23 + 19 + 24 + 46 + 15 + 21) / 8 = 28.
Step 2: Calculate the standard deviation of the data set.
To find the standard deviation, you need to calculate the difference between each data point and the mean, square those differences, and then find the average of the squared differences. Finally, take the square root of the average.
Standard Deviation = √(((14-28)^2 + (26-28)^2 + (23-28)^2 + (19-28)^2 + (24-28)^2 + (46-28)^2 + (15-28)^2 + (21-28)^2) / 8) ≈ 9.984.
Step 3: Calculate the Z-score for each data point.
The Z-score for a data point represents how many standard deviations away it is from the mean. The formula to calculate the Z-score is (data point - mean) / standard deviation.
Z-score for each data point:
Z1 = (14-28) / 9.984 ≈ -1.404
Z2 = (26-28) / 9.984 ≈ -0.200
Z3 = (23-28) / 9.984 ≈ -0.502
Z4 = (19-28) / 9.984 ≈ -0.902
Z5 = (24-28) / 9.984 ≈ -0.401
Z6 = (46-28) / 9.984 ≈ 1.802
Z7 = (15-28) / 9.984 ≈ -1.301
Z8 = (21-28) / 9.984 ≈ -0.702
Step 4: Identify the outliers.
In general, a data point is considered an outlier if its Z-score is less than -3 or greater than 3. However, this threshold can vary depending on the specific context. For simplicity, we'll consider any Z-score less than -2 or greater than 2 as an outlier.
From the Z-scores calculated above, Z6 ≈ 1.802 is greater than 2, so it can be considered as the outlier in the given data set.
b. Finding the mean with the outlier:
Since we have identified the outlier as 46, we include it in the calculation.
Mean (with the outlier) = (14 + 26 + 23 + 19 + 24 + 46 + 15 + 21) / 8 = 28.
c. Finding the mean without the outlier:
To find the mean without the outlier, we exclude the outlier from the calculation.
Mean (without the outlier) = (14 + 26 + 23 + 19 + 24 + 15 + 21) / 7 ≈ 20.857.