Let's go through the steps to calculate each of the statistical measures for the dataset:

Dataset: 7, 9, 5, 4, 6, 8, 5, 3, 7, 9, 4, 3, 7, 6, 3

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1. Mean

The mean is the average of all the values in the dataset. It is calculated as:

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

$\text{Sum of all values} = 7 + 9 + 5 + 4 + 6 + 8 + 5 + 3 + 7 + 9 + 4 + 3 + 7 + 6 + 3 = 81$

$\text{Number of values} = 15$

$\text{Mean} = \frac{81}{15} = 5.4$

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2. Median

The median is the middle value when the data is ordered in increasing order. If the number of values is odd, it is the middle number; if even, it is the average of the two middle numbers.

Ordered dataset: 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 9, 9

Since there are 15 values (odd), the median is the 8th value in the ordered list:

$\text{Median} = 6$

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3. Mode

The mode is the number that appears most frequently in the dataset.

Ordered dataset: 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 9, 9

The mode is the value that occurs the most. Here, the number 3 occurs 3 times, and the number 7 also occurs 3 times.

$\text{Mode} = 3 \text{ and } 7 \quad (\text{This is bimodal.})$

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4. Population Variance and Standard Deviation

• Population Variance ($\sigma^2$) is calculated as:

$\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$

Where $x_i$ represents each data point, $\mu$ is the mean, and $N$ is the number of data points.

Let's compute each step:

1. Mean ($\mu$) = 5.4
2. Compute $(x_i - \mu)^2$ for each value:

$(7 - 5.4)^2 = 2.56,\quad (9 - 5.4)^2 = 12.96,\quad (5 - 5.4)^2 = 0.16,\quad (4 - 5.4)^2 = 1.96$ $(6 - 5.4)^2 = 0.36,\quad (8 - 5.4)^2 = 6.76,\quad (5 - 5.4)^2 = 0.16,\quad (3 - 5.4)^2 = 5.76$ $(7 - 5.4)^2 = 2.56,\quad (9 - 5.4)^2 = 12.96,\quad (4 - 5.4)^2 = 1.96,\quad (3 - 5.4)^2 = 5.76$ $(7 - 5.4)^2 = 2.56,\quad (6 - 5.4)^2 = 0.36,\quad (3 - 5.4)^2 = 5.76$

Now, sum up the squared differences:

$\sum (x_i - \mu)^2 = 2.56 + 12.96 + 0.16 + 1.96 + 0.36 + 6.76 + 0.16 + 5.76 + 2.56 + 12.96 + 1.96 + 5.76 + 2.56 + 0.36 + 5.76 = 67.36$

Now, divide by $N = 15$:

$\sigma^2 = \frac{67.36}{15} = 4.49$

• Population Standard Deviation ($\sigma$) is the square root of the population variance:

$\sigma = \sqrt{4.49} \approx 2.12$

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5. Range

The range is the difference between the maximum and minimum values in the dataset.

$\text{Range} = \text{Max value} - \text{Min value} = 9 - 3 = 6$

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Summary of Results:

• Mean: 5.4
• Median: 6
• Mode: 3 and 7 (bimodal)
• Population Variance: 4.49
• Population Standard Deviation: 2.12
• Range: 6

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Would you like more details or have any questions on this? Here are some related questions for further practice:

1. How does the range help in understanding the variability of a dataset?
2. What would change in the results if we were calculating sample variance instead of population variance?
3. How do the mean, median, and mode relate to each other in skewed datasets?
4. How can outliers affect the mean and variance?
5. What are the differences between population and sample standard deviation?

Tip: The median is less sensitive to outliers than the mean. In datasets with extreme values, the median may provide a better representation of central tendency.