Question

For the datasets. Use technology to find the following values: (a) The mean and the standard deviation. (b) The five number summary. 10,11,13,14,14,17,18,20,21,25,28

Short answer

The Mean is 17.36, Standard Deviation is 5.53. The Five-Number Summary is {Min=10, Q1=13, Median=17, Q3=21, Max=28}

Step-by-step solution

Calculate the Mean

To calculate the mean, add up all the values in the dataset and divide by the number of data points. The dataset contains the following numbers: 10,11,13,14,14,17,18,20,21,25,28. The sum of these numbers is 191 and there are 11 numbers, so the mean is \(\frac{191}{11} = 17.36\)

Calculate the Standard Deviation

The standard deviation is calculated by taking the square root of the variance. The variance is the average of the squared differences from the Mean. First, subtract the mean from each number in the dataset and then square the result. The squared differences are: 54.39, 40.33, 19.21, 11.45, 11.45, 0.13, 0.41, 7.11, 13.28, 58.64, 113.06. The variance is the average of these values: \(\frac{\sum{Squared Differences}}{N-1}\) = 30.63. The standard deviation is the square root of the Variance: \(\sqrt{30.63} = 5.53\)

Compute the Five-Number Summary

The five-number summary includes the minimum value (10), the first quartile (Q1) which is the median of the first half of the data (13), the median (Q2) which is middle value when the data is ordered from least to greatest (17), the third quartile (Q3) which is the median of the second half of the data (21), and the maximum value (28).

Key concepts

Mean Calculation

When we talk about the mean of a dataset, we're referring to what is commonly known as the average. The process of finding the mean is quite straightforward. To calculate it, you simply add up all the numbers in the dataset and then divide that sum by the total number of data points in the set. For example, with the given dataset (10, 11, 13, 14, 14, 17, 18, 20, 21, 25, 28), you would add these 11 numbers to get a sum of 191. Dividing this sum by 11, the number of observations, yields a mean of \(\frac{191}{11} = 17.36\). The mean is a crucial measure because it introduces the concept of the center of a dataset. However, it's sensitive to outliers, meaning that a very high or very low value can significantly impact the mean.

An essential tip to remember when calculating the mean is to ensure that all data points are accounted for and that the dataset is free of errors. This will help maintain accuracy in your calculations, which is crucial for descriptive statistics.

Standard Deviation Calculation

Standard deviation is a statistic that measures the dispersion of a dataset relative to its mean. It's a useful tool for understanding how spread out the data is. Calculating the standard deviation requires a few steps. First, you need to calculate the variance, which involves finding the mean (as previously discussed) and then measuring how far each data point is from that mean. This distance is squared for each data point, and then these squared distances are averaged, but with one adjustment – we divide by the number of data points minus one (N-1) when we calculate this average. This is known as Bessel's correction, used to provide a better estimate of the population standard deviation when dealing with a sample.

In the given dataset, after squaring the differences between each data point and the mean, and averaging those, we have a variance of \(30.63\). To find the standard deviation, we take the square root of the variance, resulting in \(\sqrt{30.63} = 5.53\). A larger standard deviation indicates a greater spread of the data points from the mean. Remember, it's pivotal to square the differences to avoid negative values canceling out positive ones, which would occur if we just took the plain differences.

Five Number Summary

The five number summary includes five key data points that provide a comprehensive overview of a dataset. They are the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum values. These values divide the dataset into quarters, providing a clear picture of the distribution. For the dataset in question, here are the steps to identify these points:

• The minimum value is simply the smallest number in the set: 10.
• Q1 is the median of the lower half of the dataset: 13.
• The median (Q2) is the value that lies in the middle when you arrange the data in ascending order: 17.
• Q3 is the median of the upper half of the dataset: 21.
• The maximum value is the largest number in the set: 28.

Together, these five numbers form the so-called 'box' in a box-and-whisker plot, which is a visual representation of the five number summary. They reveal the range of your data, where the middle pack lies, and if there are any potential outliers on either end of the dataset.

Variance Calculation

Variance is a measure of how much the numbers in a dataset vary from the mean and and from each other. It represents the average of the squared differences between each data point and the mean. Here's a simple step-by-step approach to calculate it for the given dataset:

1. Calculate the mean of the dataset.
2. Subtract the mean from each data point and square the result to find the squared differences.
3. Add all the squared differences together.
4. Divide this sum by the number of data points minus one (N-1) to account for Bessel's correction.

The result is the variance of the dataset, which in our case is 30.63. Since the variance uses squared units, it’s not in the same units as the data points, and hence we often use the standard deviation (the square root of variance) to interpret the spread more intuitively. However, understanding variance is crucial because it lays the groundwork for various other statistical concepts and is the basis for the standard deviation.