Question

In an office, the employer notices his employees spend more time drinking coffee than working. He counts the number of coffee breaks each of his seven employees takes in the course of a day. The data are \(1,1,2,2,3,5\), and 7 Find the mean, variance, standard deviation and the median number of coffee breaks a day.

Short answer

The mean number of coffee breaks per day is \(3\), the variance is \(\frac{30}{7}\), the standard deviation is approximately \(1.95\), and the median is \(2\).

Step-by-step solution

Find the Mean of the Dataset

To find the mean, we sum all the numbers within the dataset and divide by the total number of data points. In this case, there are seven data points: Mean = \(\frac{1 + 1 + 2 + 2 + 3 + 5 + 7}{7}\) Mean = \(\frac{21}{7}\) Mean = \(3\) The mean number of coffee breaks per day is \(3\).

Calculate the Variance of the Dataset

To calculate the variance, we need to find the average of the squared differences from the mean. First, we find the difference between each data point and the mean, and then square the differences: 1. (1 - 3)² = 4 2. (1 - 3)² = 4 3. (2 - 3)² = 1 4. (2 - 3)² = 1 5. (3 - 3)² = 0 6. (5 - 3)² = 4 7. (7 - 3)² = 16 Next, add the squared differences and divide by the total number of data points: Variance = \(\frac{4 + 4 + 1 + 1 + 0 + 4 + 16}{7}\) Variance = \(\frac{30}{7}\) The variance of the dataset is \(\frac{30}{7}\).

Find the Standard Deviation of the Dataset

The standard deviation is the square root of the variance: Standard Deviation = \(\sqrt{\frac{30}{7}}\) Standard Deviation ≈ \(1.95\) The standard deviation of the dataset is approximately \(1.95\).

Calculate the Median of the Dataset

To find the median, we must first arrange the data points in ascending order: 1, 1, 2, 2, 3, 5, 7 The dataset has an odd number of values, so the median is the middle value: Median = \(2\) The median number of coffee breaks per day is \(2\). From this analysis, we can conclude that the mean number of coffee breaks per day is \(3\), the variance is \(\frac{30}{7}\), the standard deviation is approximately \(1.95\), and the median is \(2\).

Key concepts

Mean

The mean, often called the average, gives us the central value of a dataset. To calculate the mean, add all the numbers together and then divide by the total count of numbers. This statistic is helpful to understand what a typical data point might be.

For the dataset representing coffee breaks:

• Sum of data: \(1 + 1 + 2 + 2 + 3 + 5 + 7 = 21\)
• Count of data points: 7
• Mean: \(\frac{21}{7} = 3\)

Thus, employees take an average of 3 coffee breaks per day. This gives us a quick insight into daily habits.

Variance

Variance measures how spread out the numbers in a dataset are. It provides insight into the variability of the data points compared to the mean. A higher variance indicates more spread, while a lower variance shows that the numbers are closer to the mean.

Calculating variance involves these steps:

• Find the difference between each data point and the mean \((3)\).
• Square each of these differences to eliminate negative values.
• Calculate the average of these squared differences.

For our data:

• Squared differences: \((1 - 3)^2, (1 - 3)^2, (2 - 3)^2, \ldots\)
• Sum of squared differences: 30
• Variance: \(\frac{30}{7}\)

The variance of \(\frac{30}{7}\) shows moderate variability in coffee breaks.

Standard Deviation

Standard deviation is a measure that indicates the extent of deviation for a group as a whole. It's simply the square root of the variance. This measurement makes it easier to understand data variability in the same unit as the data itself.

To compute the standard deviation:

• Identify the variance: \(\frac{30}{7}\).
• Take the square root: \(\sqrt{\frac{30}{7}} \approx 1.95\).

The standard deviation of approximately 1.95 informs us about the average distance of each data point from the mean. A smaller standard deviation implies that data points are closer to the mean.

Median

The median is the middle value in a dataset when the numbers are arranged in order. It serves as a valuable indicator of the dataset's center, especially when dealing with outliers or skewed data that might distort the mean.

To find the median in our example:

• Arrange the data: 1, 1, 2, 2, 3, 5, 7
• Since there are 7 values, the middle one is the 4th value.
• Median: 2

The median is 2, showing that more than half of the employees take 2 or fewer coffee breaks, a number less affected by extreme values like 7.