To calculate the mean, median, mode, variance, and standard deviation for the given population data set $4, 5, 6, 6, 7, 4, 6, 4, 3, 5$, let's proceed step by step.

Step 1: Mean (Average)

The mean of a population is the sum of all values divided by the number of values.

$\text{Mean} = \frac{\sum x_i}{N}$ Where $\sum x_i$ is the sum of all the data points, and $N$ is the number of data points.

Sum of the values: $4 + 5 + 6 + 6 + 7 + 4 + 6 + 4 + 3 + 5 = 50$ Number of values ($N$) = 10

$\text{Mean} = \frac{50}{10} = 5$

Step 2: Median

The median is the middle value of the dataset when it is ordered. If the dataset has an even number of values, the median is the average of the two middle values.

Ordered data set: $3, 4, 4, 4, 5, 5, 6, 6, 6, 7$

Since there are 10 data points (even number), the median is the average of the 5th and 6th values.

The 5th and 6th values are 5 and 5, respectively.

$\text{Median} = \frac{5 + 5}{2} = 5$

Step 3: Mode

The mode is the value that occurs most frequently in the dataset.

In this case, the frequency of each value:

• 3 occurs 1 time
• 4 occurs 3 times
• 5 occurs 2 times
• 6 occurs 3 times
• 7 occurs 1 time

The values 4 and 6 both appear 3 times, which makes them bimodal.

$\text{Mode} = 4, 6$

Step 4: Variance

Variance measures the average squared deviation from the mean. For a population, the formula is:

$\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}$ Where:

• $x_i$ represents each data point
• $\mu$ is the population mean
• $N$ is the number of data points

First, we calculate the squared deviations from the mean (5):

$(4 - 5)^2 = 1, \quad (5 - 5)^2 = 0, \quad (6 - 5)^2 = 1, \quad (6 - 5)^2 = 1$ $(7 - 5)^2 = 4, \quad (4 - 5)^2 = 1, \quad (6 - 5)^2 = 1, \quad (4 - 5)^2 = 1$ $(3 - 5)^2 = 4, \quad (5 - 5)^2 = 0$

Sum of squared deviations:

$1 + 0 + 1 + 1 + 4 + 1 + 1 + 1 + 4 + 0 = 14$

Now, divide by the number of data points (N = 10):

$\text{Variance} = \frac{14}{10} = 1.4$

Step 5: Standard Deviation

The standard deviation is the square root of the variance.

$\text{Standard Deviation} = \sqrt{1.4} \approx 1.18$

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Summary of Results:

• Mean: 5
• Median: 5
• Mode: 4, 6 (bimodal)
• Variance: 1.4
• Standard Deviation: 1.18

Would you like a more detailed explanation of any of these calculations, or do you have any other questions on this topic?

Here are some related questions you might find interesting:

1. How does the variance differ between a sample and a population?
2. What does it mean for a dataset to be bimodal, and how does it affect the mean and median?
3. How do you interpret the standard deviation in the context of this dataset?
4. What is the significance of having multiple modes in a dataset?
5. How does changing the value of a data point impact the variance and standard deviation?

Tip: To better understand variability, try calculating the range (difference between the maximum and minimum values), which gives a quick sense of spread in the data.