Problem 54 - Thinking Mathematically

Question

Use the data \(0,1,3,4,4,6\). Add 2 to each of the numbers. How does this affect the mean? How does this affect the standard deviation?

Step-by-Step Solution

Answer

Adding 2 to each number in the data set increases the mean from 3 to 5, but doesn't affect the standard deviation, which remains the same at 2.

Step 1: Calculate the Mean of the Original Data

Firstly, add up all the numbers in the data set and then divide by how many numbers there are. So, sum of \(0,1,3,4,4,6\) is \(18\). As there are 6 numbers, the mean is \(\frac{18}{6}=3\)

Step 2: Calculate the Standard Deviation of the Original Data

Subtract the mean from each of the numbers to get the deviation, square these deviations, add them all together and divide by the total number of numbers. Here deviations are \[-3, -2, 0, 1, 1, 3\]. The squares of deviations are \[9, 4, 0, 1, 1, 9\]. The sum of squares is \(24\) and as there are 6 numbers, the variance is \(\frac{24}{6}=4\). The standard deviation is the square root of variance, \(\sqrt{4}=2\)

Step 3: Add 2 to Each Number and Calculate the New Mean and Standard Deviation

The new data set after adding two to each number is \[2, 3, 5, 6, 6, 8\]. The sum of the new numbers is \(30\) and the new mean is \(\frac{30}{6}=5\). The new deviations are \[-3, -2, 0, 1, 1, 3\] same as before. So the new variance calculated same as step 2 is still \(4\) and hence the new standard deviation remains \(\sqrt{4}=2\)

Key Concepts

Understanding the Data Set

In statistics, a data set is a collection of numbers that we aim to analyze using various statistical methods. For instance, consider the numbers:

• Original data: \(0, 1, 3, 4, 4, 6\)
• Modified data (after adding 2 to each number): \(2, 3, 5, 6, 6, 8\)

When working with data sets, we typically perform calculations to understand the distribution and tendencies within the data. Identifying patterns or shifts in a data set helps us make informed conclusions.

Mean Calculation Explained

The mean, or average, represents a central value of a data set. To calculate the mean, sum all the data points and divide by the number of points.
For our original data set \(0, 1, 3, 4, 4, 6\), the mean is calculated as follows:

• Sum of numbers: \(0 + 1 + 3 + 4 + 4 + 6 = 18\)
• Count of numbers: 6
• Mean: \(\frac{18}{6} = 3\)

In the modified data set \(2, 3, 5, 6, 6, 8\), the sum is 30. Thus, the mean becomes \(\frac{30}{6} = 5\). Adding a constant shifts the mean by that constant amount.

Standard Deviation Calculation and Its Role

Standard deviation quantifies the amount of variation or dispersion in a data set. It tells us how spread out the numbers are around the mean. To calculate the standard deviation:

• Subtract the mean from each data point to get the deviations.
• Square each of these deviations.
• Find the average of these squared deviations (known as variance).
• Take the square root of the variance to get the standard deviation.

In our example, the variance of the original data set was 4, and thus the standard deviation was \(\sqrt{4} = 2\). This measurement helps you understand the spread of the data points.

Explaining Variance

Variance is closely related to standard deviation and represents the average of the squared differences from the mean. It provides insight into the data's variability:

• Higher variance means data points are spread out over a wider range.
• Lower variance indicates data points are closer to the mean.

In our data example, both the original and modified data sets have a variance of 4. This uniformity confirms that adding a constant affects only the position of the data set, not its variability.

Effects of Adding a Constant

When a constant is added to every entry in a data set, it impacts the mean but not the spread. The reason is straightforward:

• Adding a constant increases the total sum of the data set, thus increasing the mean by the same constant value.
• However, the amount each data point differs from the mean does not change, keeping the variance and standard deviation identical.

In our scenario, adding 2 raised the mean from 3 to 5, but the standard deviation remained 2. This illustrates how such operations affect central tendency without affecting data dispersion.