Question 1184765: For the following sample of scores, find the mean, median, mode, range and ss, variance, and standard deviation. What is the shape of this distribution? What is the best way to report average in this distribution?
x f
18 1
17 2
16 4
15 5
14 4
13 2
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2 1
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **1. Calculate the Mean:**
* Multiply each score (x) by its frequency (f) and sum the results:
(18 * 1) + (17 * 2) + (16 * 4) + (15 * 5) + (14 * 4) + (13 * 2) + (2 * 1) = 295
* Sum the frequencies: 1 + 2 + 4 + 5 + 4 + 2 + 1 = 19
* Divide the sum of the products by the sum of the frequencies:
Mean = 295 / 19 = 15.53
**2. Determine the Median:**
* The median is the middle score when the scores are arranged in order. Since there are 19 scores, the median is the 10th score.
* Listing the scores in order with their frequencies, we find the 10th score is 15.
Therefore, the median is 15.
**3. Identify the Mode:**
* The mode is the score with the highest frequency. In this case, the score 15 has the highest frequency (5).
Therefore, the mode is 15.
**4. Calculate the Range:**
* Range = Highest Score - Lowest Score = 18 - 2 = 16
**5. Calculate the Sum of Squares (SS):**
* Subtract the mean from each score, square the result, multiply by the frequency, and sum the products:
SS = (1 * (18 - 15.53)^2) + (2 * (17 - 15.53)^2) + (4 * (16 - 15.53)^2) + (5 * (15 - 15.53)^2) + (4 * (14 - 15.53)^2) + (2 * (13 - 15.53)^2) + (1 * (2 - 15.53)^2)
SS = 6.10 + 4.18 + 0.89 + 1.37 + 9.24 + 13.17 + 183.33 = 218.28
**6. Calculate the Variance:**
* Divide the sum of squares (SS) by the sum of the frequencies minus 1:
Variance = SS / (n - 1) = 218.28 / (19 - 1) = 12.24
**7. Calculate the Standard Deviation:**
* Take the square root of the variance:
Standard Deviation = sqrt(Variance) = sqrt(12.24) = 3.50
**Shape of the Distribution:**
* The distribution appears to be slightly skewed to the left (negatively skewed) due to the outlier (2). The majority of the scores are clustered around the mean, but the outlier pulls the mean slightly to the left.
**Best Way to Report the Average:**
* Due to the presence of the outlier, the median (15) might be a better measure of central tendency than the mean (15.53) in this case. The median is less affected by extreme values.
**Summary:**
* Mean = 15.53
* Median = 15
* Mode = 15
* Range = 16
* SS = 218.28
* Variance = 12.24
* Standard Deviation = 3.50
* Shape = Slightly Skewed Left (Negative Skew)
* Best Average = Median
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