Question

Use these \(n=15\) measurements to answer the questions in Exercises 11 and 12. 53,61,58,56,58,60,54,54,62,58,60,58,56,56,58 Calculate \(\bar{x}, m,\) and the mode.

Short answer

Based on the given dataset, find the mean, median, and mode.

Step-by-step solution

Calculate the mean \(\bar{x}\)

To calculate the mean, first sum all the measurements in the dataset: \(53+61+58+56+58+60+54+54+62+58+60+58+56+56+58 = 862\) Then, divide this sum by the total number of measurements (\(n=15\)): Mean, \(\bar{x} = \frac{862}{15} = 57.467\)

Calculate the median \(m\)

To calculate the median, first sort the measurements in ascending order: \(53, 54, 54, 56, 56, 56, 58, 58, 58, 58, 60, 60, 61, 62\) Since there is an odd number of measurements (\(n=15\)), the median will be the middle value, located at position \(\frac{n+1}{2} = \frac{15+1}{2} = 8\). In the sorted list, the value at the 8th position is: Median, \(m = 58\)

Calculate the mode

To calculate the mode, find the measurement with the highest frequency in the dataset. The measurements and their frequencies are: - 53: 1 - 54: 2 - 56: 3 - 58: 4 - 60: 2 - 61: 1 - 62: 1 The measurement with the highest frequency is 58, appearing 4 times. Mode = 58 The mean \(\bar{x}\) is \(57.467\), the median \(m\) is \(58\), and the mode is \(58\).

Key concepts

Mean Calculation

The mean, often referred to as the average, serves as a central measure of a data set. It gives us a quick snapshot of the "central location" of the numbers you have, by providing a single value that represents the entire data set. To calculate the mean, you first need to add up all the numbers in your list. For our dataset:

• The numbers are: 53, 61, 58, 56, 58, 60, 54, 54, 62, 58, 60, 58, 56, 56, and 58.
• Add them together, resulting in a total sum of 862.

Next, divide this sum by the number of measurements, which is 15 in this case. The formula for the mean \( \bar{x} \) is: \[ \bar{x} = \frac{\text{sum of all measurements}}{\text{total number of measurements}} = \frac{862}{15} = 57.467 \]This means the average of this data set is approximately 57.467.

Median Calculation

The median provides another form of central tendency by identifying the middle value of a data set when the numbers are arranged in order from smallest to largest. It helps to illustrate the typical value in data, less affected by extreme values (outliers) compared to the mean.
To find the median, you need to first arrange your measurements in increasing order:

• Ordered list: 53, 54, 54, 56, 56, 56, 58, 58, 58, 58, 60, 60, 61, 62.

With 15 observations, which is an odd number, the median is located exactly in the middle. This is calculated using the formula:\[ \text{Median position} = \frac{n+1}{2} = \frac{15+1}{2} = 8 \]Hence, in our list, the 8th position is taken up by the value 58, making the median 58. This reveals that when ordered, the middle of the dataset tends to be around 58.

Mode Calculation

The mode is a measure of central tendency that identifies the most frequently occurring value within a dataset. It provides insights into the commonality within the dataset.
To find the mode, you look for the number that appears most frequently:

• In our data: 53 appears 1 time, 54 appears 2 times, 56 appears 3 times, 58 appears 4 times, 60 appears 2 times, and both 61 and 62 appear 1 time each.

Clearly, the number 58 appears the most often in the dataset, with a frequency of 4 times. Thus, the mode of this data is 58.
This indicates that in this measured group, 58 was the most common value among all measured values.