Let's go step by step to calculate the required values for the data set:
$118, 136, 123, 145, 167, 152, 171, 179, 168, 189, 128, 156, 167, 145, 170, 181, 117, 145, 122, 135$

a) Mean:

The mean is the sum of all the numbers divided by the number of values in the data set.

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

First, sum all the numbers: $118 + 136 + 123 + 145 + 167 + 152 + 171 + 179 + 168 + 189 + 128 + 156 + 167 + 145 + 170 + 181 + 117 + 145 + 122 + 135 = 3145$

Number of data points $N = 20$.

$\text{Mean} = \frac{3145}{20} = 157.25$

b) Median:

The median is the middle number when the data is sorted in ascending order.

1. Sort the data in ascending order: $117, 118, 122, 123, 128, 135, 136, 145, 145, 145, 152, 156, 167, 167, 168, 170, 171, 179, 181, 189$

2. Since there are 20 data points (even number), the median is the average of the 10th and 11th numbers in the sorted list.

The 10th and 11th values are: $145 \quad \text{and} \quad 152$

The median is: $\text{Median} = \frac{145 + 152}{2} = \frac{297}{2} = 148.5$

c) Mode:

The mode is the value that appears most frequently in the data set.

Looking at the data, the number 145 appears 3 times, which is more frequent than any other number.

$\text{Mode} = 145$

d) Variance:

Variance measures how far the numbers in the data set are spread out from the mean.

$\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}$ Where $x_i$ represents each data point, and $\mu$ is the mean.

1. Subtract the mean from each data point and square the result.
2. Sum all the squared differences.
3. Divide the sum by the number of data points.

I'll calculate the squared differences from the mean and their sum:

$(118 - 157.25)^2 = 1536.0625, \quad (136 - 157.25)^2 = 448.5625, \quad (123 - 157.25)^2 = 1163.0625, \quad \ldots$

Summing the squared differences: $1536.0625 + 448.5625 + 1163.0625 + \ldots = 12904.75$

Now, divide by the number of data points (20): $\text{Variance} = \frac{12904.75}{20} = 645.2375$

e) Standard Deviation:

The standard deviation is the square root of the variance.

$\text{Standard Deviation} = \sqrt{645.2375} \approx 25.42$

Summary:

• Mean: 157.25
• Median: 148.5
• Mode: 145
• Variance: 645.24
• Standard Deviation: 25.42

Would you like further details on any of these calculations? Let me know if you need anything clarified.

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Further Exploration:

1. How does changing one value in the data set affect the mean and variance?
2. What is the effect of an outlier on the standard deviation?
3. How would the results change if the data set were skewed?
4. What are the differences between sample and population variance and standard deviation?
5. How can you interpret the spread of data using variance and standard deviation?

Tip: When analyzing data, remember that the mean provides an average, the median gives a central value, and the mode reveals the most frequent values. Together, they help you understand different aspects of the data.