Question

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? $$ 53,61,58,56,58,60,54,54,62,58,60,58,56,56,58 $$

Short answer

Answer: The range is approximately 3.42 standard deviations.

Step-by-step solution

Calculate the mean

To calculate the mean, add all the given values and divide the sum by the total number of values: $$ \text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of values}} $$ $$ \text{Mean} = \frac{53+61+58+56+58+60+54+54+62+58+60+58+56+56+58}{15} $$ $$ \text{Mean} = \frac{861}{15} $$ $$ \text{Mean} ≈ 57.4 $$

Calculate the standard deviation

To calculate the standard deviation, follow these steps: a. Subtract the mean from each value and square the result. b. Calculate the mean of the squared differences. c. Take the square root of the mean of the squared differences. The formula for the standard deviation is: $$ \text{Standard Deviation} = \sqrt{\frac{\sum_{i}^{N} (x_i - \text{Mean})^2}{N}} $$ $$ \text{Standard Deviation} ≈ \sqrt{\frac{(53-57.4)^2+(61-57.4)^2+...+(58-57.4)^2}{15}} $$ $$ \text{Standard Deviation} ≈ 2.63 $$

Calculate the range

To calculate the range, subtract the minimum value from the maximum value in the given data: $$ \text{Range} = \text{Maximum value} - \text{Minimum value} $$ $$ \text{Range} = 62 - 53 $$ $$ \text{Range} = 9 $$

Find how many standard deviations the range is

To find out how many standard deviations the range is, divide the range by the standard deviation: $$ \text{Number of Standard Deviations} = \frac{\text{Range}}{\text{Standard Deviation}} $$ $$ \text{Number of Standard Deviations} = \frac{9}{2.63} $$ $$ \text{Number of Standard Deviations} ≈ 3.42 $$ The range is approximately 3.42 standard deviations.

Key concepts

Understanding the Mean

The mean, often referred to as the average, is a measure of the central tendency of a data set. To find the mean, sum up all the numbers in the data set and then divide by the total count of numbers. This gives a single number that represents the "center" of the data set.

For example, let's consider the data set:

• 53, 61, 58, 56, 58, 60, 54, 54, 62, 58, 60, 58, 56, 56, 58

Adding up these values gives us 861. There are 15 numbers in the list. Thus, the mean is calculated as \( \frac{861}{15} \approx 57.4 \).

The mean is useful as it provides a simple summary of the entire data set with just one value. It helps to quickly understand where the bulk of the data lies.

Exploring the Standard Deviation

The standard deviation is a statistic that tells us how spread out the data values are. It gauges the "variance" in the data.To compute the standard deviation, follow these steps:

• Subtract the mean from each number to find the deviation for each data point.
• Square these deviations to eliminate negative values.
• Find the average of these squared deviations, and then take the square root of this average.

For our example, the mean is 57.4. So, we find each deviation, square it, average those squares and finally, take the square root to get the standard deviation, which is approximately \( 2.63 \).

A smaller standard deviation means the data points are close to the mean, whereas a larger one indicates more spread.

Calculating the Range

The range is a simple measure of dispersion, reflecting the difference between the largest and smallest values in the data set. Finding the range is straightforward:

For our data set, the maximum value is 62 and the minimum is 53. The range is calculated as follows:

• Range \( = 62 - 53 = 9 \)

The range can offer a quick glimpse into the spread of the data points, showing us how far apart the highest and lowest values are.

Despite being informative, the range does not consider the distribution or the intermediate values, which makes it a more basic metric. It provides a limited view of the data's variability.

Delving into Data Analysis

Data analysis involves evaluating data sets to draw conclusions about the information they contain. Key statistical measures like the mean, standard deviation, and range are often used.

• The mean gives us insight into the average value, summarizing the entire data set.
• The standard deviation indicates how much each number varies from that average.
• The range gives a basic sense of all observed values' extremes.

Using these tools, you can assess patterns and differences among data points. For instance, in this dataset, the mean helps locate the center, the standard deviation shows data consistency, and the range offers a picture of the data's spread.
One can also compare how many times the range fits into the standard deviation, which is useful for understanding data deviation relative to extreme values. In this case, the range is approximately 3.42 times the standard deviation, indicating significant spread.