- The mean is the average value found by summing all observations and dividing by the total count.
- The sum of the sit-up counts is \(24 + 26 + 27 + 32 + 25 + 25 + 30 = 189\).
- The total number of students is \(7\).
- Dividing the sum by the count gives the mean: \(\frac{189}{7} = \boxed{27}\).
Explanation
- Analyze the data and define the mean
The provided data consists of the number of sit-ups performed by Manny and his classmates. The dataset is given as: \(\{24, 26, 27, 32, 25, 25, 30\}\). There are a total of \(n = 7\) observations. The mean of a data set is defined as the arithmetic average, calculated by dividing the sum of all observations by the number of observations.
- Calculate the sum of the data points
To find the mean, we first calculate the sum (\(\Sigma\)) of all the sit-up counts in the list:
\[\Sigma = 24 + 26 + 27 + 32 + 25 + 25 + 30\]
Summing these step by step:
\(24 + 26 = 50\)
\(50 + 27 = 77\)
\(77 + 32 = 109\)
\(109 + 25 = 134\)
\(134 + 25 = 159\)
\(159 + 30 = 189\)
The total sum of all sit-ups is \(189\).
- Determine the mean of the data set
Now that we have the total sum, we divide it by the total number of observations, \(n = 7\), to find the mean (\(\bar{x}\)):
\[\bar{x} = \frac{\Sigma}{n} = \frac{189}{7}\]
Performing the division:
\(189 \div 7 = 27\)
Therefore, the mean number of sit-ups for the students is \(27\).
Examples
In real-life scenarios, calculating the mean helps summarize a set of numerical data into a single, representative value. For instance, if you want to know the average daily steps you walked over a week, you would sum the total steps taken each day and divide by the number of days, similar to how we calculated the average number of sit-ups here. This is a fundamental technique used in everything from tracking personal fitness goals to analyzing academic performance across a class. For a dataset \(\{x_1, x_2, \dots, x_n\}\), the mean \(\bar{x}\) is calculated as \(\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\).