The numbers of regular season wins for 10 football teams in a given season are given below. Determine the range, mean, variance, and standard deviation of the population data set. 2,7, 15, 4, 14, 6, 13 , 8, 3, 10 The range is 13 (Simplify your answer.) The population mean is (Simplify your answer. Round to the nearest tenth as needed.)
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
![### Determining Statistical Values from a Population Data Set
The numbers of regular season wins for 10 football teams in a given season are provided below. The goal is to determine the range, mean, variance, and standard deviation of this population data set.
Given data set:
\[2, 7, 15, 4, 14, 6, 13, 8, 3, 10\]
**Step-by-Step Solution:**
1. **Range:**
- The range is the difference between the maximum and minimum values in the dataset.
- From the given data:
\[
\text{Maximum value} = 15, \quad \text{Minimum value} = 2
\]
- Therefore, the range is:
\[
\text{Range} = 15 - 2 = 13
\]
- The range is \(13\)
2. **Mean (Average):**
- The mean is the sum of all values divided by the number of values.
- Sum of the data set:
\[
2 + 7 + 15 + 4 + 14 + 6 + 13 + 8 + 3 + 10 = 82
\]
- Number of values:
\[
n = 10
\]
- Therefore, the mean is:
\[
\text{Mean} = \frac{82}{10} = 8.2
\]
- The population mean is \(8.2\)
3. **Variance:**
- Variance is the average of the squared differences from the mean.
- Steps to calculate variance:
1. Calculate the mean (already found to be 8.2).
2. Subtract the mean from each data value and square the result:
\[
(2 - 8.2)^2, (7 - 8.2)^2, (15 - 8.2)^2, (4 - 8.2)^2, (14 - 8.2)^2, (6 - 8.2)^2, (13 - 8.2)^2, (8 - 8.2)^2, (3 - 8.2)^2, (10 - 8.2)^2
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