Question

For the data sets in Exercises \(4-6,\) use the data entry method in your scientific calculator to enter the measurements. Recall the proper memories to find the mean and standard deviation. Calculate the range. The range is approximately how many standard deviations? $$ \begin{array}{llllll} 4.5 & 3.2 & 3.5 & 3.9 & 3.5 & 3.9 \\ 4.3 & 4.8 & 3.6 & 3.3 & 4.3 & 4.2 \\ 3.9 & 3.7 & 4.3 & 4.4 & 3.4 & 4.2 \\ 4.4 & 4.0 & 3.6 & 3.5 & 3.9 & 4.0 \end{array} $$

Short answer

Answer: To determine the range of the data set, find the highest and lowest measurements in the data, and then subtract the minimum value (min) from the maximum value (max). The range is equal to the difference between the max and min values.

Step-by-step solution

Input data into the scientific calculator

Use the data entry method in your scientific calculator to input the given measurements. For each calculator, the data entry process may differ, so refer to the user manual of your calculator for specific instructions.

Calculate mean and standard deviation

After inputting the data, use your calculator's built-in functions to find the mean (\(\mu\)) and standard deviation (\(\sigma\)) of the data. In most calculators, you will do this by recalling the proper memories. Again, refer to your calculator's user manual for specific steps.

Determine the range

To find the range of the data set, identify the highest and lowest measurements in the data: \(max = 4.8\) \(min = 3.2\) Then, subtract the minimum value from the maximum value: \(Range = max - min = 4.8 - 3.2 = 1.6\)

Calculate the number of standard deviations in the range

Now that we have the range, divide the range by the standard deviation calculated earlier to find out how many standard deviations the range is approximately equal to: \(\frac{Range}{\sigma} = \frac{1.6}{\sigma}\) This is the final answer. Since the value of \(\sigma\) is calculator-dependent, plug in the standard deviation calculated using your scientific calculator to get the final result.

Key concepts

Mean Calculation

The mean, often referred to as the average, is a fundamental concept in descriptive statistics. It gives us a central value for a data set. To calculate the mean, sum all the values in the set and then divide by the number of values. Consider our data set:\[\begin{array}{lllll}4.5 & 3.2 & 3.5 & 3.9 & 3.5 & 3.9\4.3 & 4.8 & 3.6 & 3.3 & 4.3 & 4.2\3.9 & 3.7 & 4.3 & 4.4 & 3.4 & 4.2\4.4 & 4.0 & 3.6 & 3.5 & 3.9 & 4.0 \end{array}\]**Steps to Calculate Mean:**

• Add all numbers together:
• Calculate the sum: \(4.5 + 3.2 + 3.5 + \ldots + 4.0\)
• Divide the total sum by the total number of values, which is 24 in this example.

Using a calculator simplifies this process and gives a precise mean value. The mean is crucial as it provides insight into the general trend of data.

Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates a wide spread of values. **Steps to Calculate Standard Deviation:**

• Find the mean of the data set.
• Subtract the mean from each data point and square the result.
• Calculate the average of these squared differences.
• Take the square root of this average to get the standard deviation.

For complex data sets, using a scientific calculator is recommended. It usually involves entering data and using built-in statistical functions to find the standard deviation quickly. Understanding standard deviation helps gauge the stability of processes or datasets, which is useful for identifying trends and estimating probabilities.

Range Calculation

The range provides a simple measure of variability in a data set. It indicates the span of data by subtracting the minimum value from the maximum value.**Steps to Calculate Range:**

• Identify the maximum value in the data set: 4.8.
• Identify the minimum value in the data set: 3.2.
• Subtract the minimum from the maximum: \(4.8 - 3.2 = 1.6\).

This gives the range of the data set. The range also helps determine how many standard deviations the data spread encompasses. By dividing the range by the standard deviation, we can assess the relative variability of the data using the standard deviation measure. This helps interpret data consistency.