Question 1184764: For the following distribution of population scores, find the mean, median, mode, range, SIQR, 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
20 1
19 2
18 5
17 3
16 1
15 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:
(20 * 1) + (19 * 2) + (18 * 5) + (17 * 3) + (16 * 1) + (15 * 1) = 246
* Sum the frequencies: 1 + 2 + 5 + 3 + 1 + 1 = 13
* Divide the sum of the products by the sum of the frequencies:
Mean = 246 / 13 = 18.92
**2. Determine the Median:**
* The median is the middle score when the scores are arranged in order. Since there are 13 scores, the median is the 7th score.
* Listing the scores in order with their frequencies, we find the 7th score is 18.
Therefore, the median is 18.
**3. Identify the Mode:**
* The mode is the score with the highest frequency. In this case, the score 18 has the highest frequency (5).
Therefore, the mode is 18.
**4. Calculate the Range:**
* Range = Highest Score - Lowest Score = 20 - 15 = 5
**5. Calculate the Semi-Interquartile Range (SIQR):**
* First, find the first quartile (Q1), which is the median of the lower half of the data. In this case, the lower half has 6 scores, so Q1 is the average of the 3rd and 4th scores: (18 + 18) / 2 = 18.
* Next, find the third quartile (Q3), which is the median of the upper half of the data. The upper half also has 6 scores, so Q3 is the average of the 10th and 11th scores: (18 + 19) / 2 = 18.5.
* SIQR = (Q3 - Q1) / 2 = (18.5 - 18) / 2 = 0.25
**6. 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 * (20 - 18.92)^2) + (2 * (19 - 18.92)^2) + (5 * (18 - 18.92)^2) + (3 * (17 - 18.92)^2) + (1 * (16 - 18.92)^2) + (1 * (15 - 18.92)^2)
SS = 1.17 + 0.01 + 4.23 + 10.65 + 8.53 + 15.37 = 40.0
**7. Calculate the Variance:**
* Divide the sum of squares (SS) by the sum of the frequencies:
Variance = SS / n = 40.0 / 13 = 3.08
**8. Calculate the Standard Deviation:**
* Take the square root of the variance:
Standard Deviation = sqrt(Variance) = sqrt(3.08) = 1.75
**Shape of the Distribution:**
* The distribution appears to be roughly symmetrical, with the majority of the scores clustered around the mean. There is a slight negative skew due to the lower scores (15 and 16) having lower frequencies.
**Best Way to Report the Average:**
* Since the distribution is roughly symmetrical, either the mean (18.92) or the median (18) would be appropriate measures of central tendency. The mode (18) could also be reported as it coincides with the median and is near the mean.
**Summary:**
* Mean = 18.92
* Median = 18
* Mode = 18
* Range = 5
* SIQR = 0.25
* SS = 40.0
* Variance = 3.08
* Standard Deviation = 1.75
* Shape = Roughly Symmetrical with Slight Negative Skew
* Best Average = Mean or Median
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