sometimes outliers can have a significant effect on statistics, and sometimes they don’t. use the following…

sometimes outliers can have a significant effect on statistics, and sometimes they don’t. use the following distribution to determine which statistics are affected by outliers. round to the nearest hundredth.\n2, 4, 5, 7, 8\n1. find the median of the distribution.\n2. find the mean of the distribution.\n3. find the standard deviation of the distribution.\n4. make a new distribution consisting of the same first four numbers, but change the 8 to 800. find the mean, median, and standard deviation.\n5. look at your answers to part d. which statistic was least changed by the outlier of 800?

sometimes outliers can have a significant effect on statistics, and sometimes they don’t. use the following distribution to determine which statistics are affected by outliers. round to the nearest hundredth.\n2, 4, 5, 7, 8\n1. find the median of the distribution.\n2. find the mean of the distribution.\n3. find the standard deviation of the distribution.\n4. make a new distribution consisting of the same first four numbers, but change the 8 to 800. find the mean, median, and standard deviation.\n5. look at your answers to part d. which statistic was least changed by the outlier of 800?

Answer

Explanation:

Step1: Find the median of 2, 4, 5, 7, 8

The data is already in ascending - order. For a set of (n = 5) (odd number of) data points, the median is the (\left(\frac{n + 1}{2}\right))-th value. (\frac{5+1}{2}=3) - rd value. So the median is 5.

Step2: Find the mean of 2, 4, 5, 7, 8

The mean (\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}), where (x_{i}) are the data - points and (n = 5). (\sum_{i=1}^{5}x_{i}=2 + 4+5 + 7+8=26), so (\bar{x}=\frac{26}{5}=5.2).

Step3: Find the standard deviation of 2, 4, 5, 7, 8

First, find the variance (s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}). ((2 - 5.2)^{2}=(-3.2)^{2}=10.24), ((4 - 5.2)^{2}=(-1.2)^{2}=1.44), ((5 - 5.2)^{2}=(-0.2)^{2}=0.04), ((7 - 5.2)^{2}=(1.8)^{2}=3.24), ((8 - 5.2)^{2}=(2.8)^{2}=7.84). (\sum_{i = 1}^{5}(x_{i}-5.2)^{2}=10.24 + 1.44+0.04 + 3.24+7.84 = 22.8). (s^{2}=\frac{22.8}{4}=5.7), and (s=\sqrt{5.7}\approx2.39).

Step4: For the new distribution 2, 4, 5, 7, 800

Median: The data in ascending - order is 2, 4, 5, 7, 800. Since (n = 5), the median is the 3 - rd value, which is 5. Mean: (\sum_{i=1}^{5}x_{i}=2 + 4+5 + 7+800=818), so (\bar{x}=\frac{818}{5}=163.6). Standard deviation: First, find the variance (s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}). ((2-163.6)^{2}=(-161.6)^{2}=26110.56), ((4 - 163.6)^{2}=(-159.6)^{2}=25472.16), ((5 - 163.6)^{2}=(-158.6)^{2}=25153.96), ((7 - 163.6)^{2}=(-156.6)^{2}=24523.56), ((800 - 163.6)^{2}=(636.4)^{2}=404904.96). (\sum_{i = 1}^{5}(x_{i}-163.6)^{2}=26110.56+25472.16+25153.96+24523.56+404904.96 = 506165.2). (s^{2}=\frac{506165.2}{4}=126541.3), and (s=\sqrt{126541.3}\approx355.73).

Step5: Compare the changes

The original median was 5, and the new median is 5. The original mean was 5.2, and the new mean is 163.6. The original standard deviation was approximately 2.39, and the new standard deviation is approximately 355.73. The median was least changed by the outlier of 800.

Answer:

  1. 5
  2. 5.2
  3. 2.39
  4. Median: 5, Mean: 163.6, Standard deviation: 355.73
  5. Median