Question

The athletic departments at 10 randomly selected U.S. universities were asked by the Equal Employment Opportunity Commission to state what percentage of their nursing scholarships were presently held by women. The responses were 5, 4, 2, 1, 1, 2, 10, 2, 3, 5.

Required:
Find the mean, median, mode, and geometric mean. Which is the most appropriate measure of central tendency? The least appropriate? Explain your answer. Is there an outlier?

Answer 1

The mean of the data is 3.5, the median is 2.5, the mode is 2, and the geometric mean is approximately 2.71. The most appropriate measure of central tendency may be the median due to the presence of an outlier, which is the value 10.

Step-by-step explanation:

The student is asked to calculate various measures of central tendency for a given set of data, which includes the mean, median, mode, and geometric mean. To find these:

1. Mean: Add all the percentages together and divide by the number of universities. (5 + 4 + 2 + 1 + 1 + 2 + 10 + 2 + 3 + 5) / 10 = 35 / 10 = 3.5
2. Median: Arrange the data in ascending order (1, 1, 2, 2, 2, 3, 4, 5, 5, 10) and find the middle number. Since there is an even number of data points, average the middle two numbers. (2 + 3) / 2 = 2.5
3. Mode: The number that occurs most frequently. In this case, it is 2.
4. Geometric Mean: Take the nth root (where n is the number of values in the dataset) of the product of the values. (1*1*2*2*2*3*4*5*5*10)^(1/10) = (4000)^(1/10) ≈ 2.71

The most appropriate measure of central tendency depends on the distribution of the data. The mean is usually most appropriate, but it is sensitive to outliers. The median is less affected by outliers and skewness. The geometric mean is helpful when dealing with rates of change or different scales. In this case, the mean may be less appropriate due to the high value of 10, which could be an outlier. The geometric mean may also be appropriate, considering the nature of percentages. The mode is least appropriate, as it does not represent the overall distribution well.

Answer 2

Answer: Mean = 3.5 , median = 2.5, mode = 2, geometric mean = 2.74

Median is the most appropriate measure of central tendency.

The least appropriate = mean

Yes there is an outlier.

Step-by-step explanation:

Given responses : 5, 4, 2, 1, 1, 2, 10, 2, 3, 5.

First arrange them in increasing order,
`\sqrt[10]{1*1*2*2*2*3*4*5*5*10 }\\\\=\sqrt[10]{24000} \approx2.74`

Its sum = 35

Mean of n observations = (Sum of observations) ÷ n

Mean = (35) ÷ 10

=3.5

Here n =10 (even)

Median = average of middle most numbers =
`(2+3)/(2)=\frac52=2.5`

Mode = most repeated number = 2 (thrice)

geometric mean =
`\sqrt[n]{x_1* x_2*.... x_n}`

10 is an outlier as it is very large as compare to other numbers.

When outlier is present in data , the median is the most appropriate measure of central tendency.

Mean affected badly by the outlier so it the least appropriate.