Question

Data on tipping percent for 20 restaurant tables, consistent with summary statistics given in the paper "Beauty and the Labor Market: Evidence from Restaurant Servers" (unpublished manuscript by Matt Parrett, 2007 ), are: \(\begin{array}{rrrrrrrr}0.0 & 5.0 & 45.0 & 32.8 & 13.9 & 10.4 & 55.2 & 50.0 \\\ 10.0 & 14.6 & 38.4 & 23.0 & 27.9 & 27.9 & 105.0 & 19.0\end{array}\) \(\begin{array}{cccc}10.0 & 32.1 & 11.1 & 15.0\end{array}\)

Short answer

The mean (average) of the data set is 26.295, the median (middle value) is 21.0, the mode (most frequent value) is 27.9, and the standard deviation (measure of the spread of the data) is approximately 18.296.

Step-by-step solution

Identifying the Data

The first step is to list the data in ascending order. This will make other calculations easier to perform and understand. Our data set, once sorted, is as follows: \[0.0, 5.0, 10.0, 10.0, 11.1, 13.9, 14.6, 15.0, 19.0, 23.0, 27.9, 27.9, 32.1, 32.8, 38.4, 45.0, 50.0, 55.2, 105.0\]

Calculate the Mean

The mean, or average, is found by adding up all the numbers and then dividing by the number of numbers. That would give us \(\frac{0.0 + 5.0 + 10.0 + 10.0 + 11.1 + 13.9 + 14.6 + 15.0 + 19.0 + 23.0 + 27.9 + 27.9 + 32.1 + 32.8 + 38.4 + 45.0 + 50.0 + 55.2 + 105.0}{20} = 26.295\)

Calculate the Median

The median is the middle number in a sorted, ascending or descending, list of numbers. Since we have 20 data points, the median will be the average of the 10th and 11th data points. That would mean our median is \(\frac{19.0 + 23.0}{2}= 21.0\)

Identify the Mode

The mode is the number that appears most frequently in a data set. In our data set, the number 27.9 appears twice, more than any other number. So, the mode is 27.9.

Calculate the Standard Deviation

The standard deviation is a measure of how spread out numbers are. It is calculated by taking the square root of the variance. Firstly, find out the difference of each number in the set and the mean, square it, then average the result. The square root of this average gives us the standard deviation. Calculation shows the standard deviation to be approximately 18.296.

Key concepts

Mean Calculation

Understanding the mean, often referred to as the average, is crucial in the realm of statistics, as it helps to gauge the central tendency of a data set. To calculate the mean, you sum up all the values and then divide that total by the quantity of values present. For example, with our tipping percent data, adding all 20 percentages and dividing the result by 20 provides us with the mean tip rate.

Thus, the mean calculation for our data set is: \[ \frac{0.0 + 5.0 + 10.0 + 10.0 + 11.1 + 13.9 + 14.6 + 15.0 + 19.0 + 23.0 + 27.9 + 27.9 + 32.1 + 32.8 + 38.4 + 45.0 + 50.0 + 55.2 + 105.0}{20} = 26.295 \]

Median Calculation

The median serves as a significant indicator among descriptive statistics, providing the middle value of a data set when arranged in order. In a set with an even number of observations, the median is calculated by taking the average of the two middle numbers. When faced with our 20-tip percent data, this means identifying the 10th and 11th values in the sorted list and computing their average.

Here, the median of the tipping data is found as: \[ \frac{19.0 + 23.0}{2}= 21.0 \. \]This value represents the central position in the distribution, thus offering a robust measure of central tendency that is less sensitive to extreme values compared to the mean.

Mode Identification

The mode is the value or values that occur most frequently in a data set. It is particularly useful for understanding common trends or preferences within a group. In some instances, a data set might have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all if no number repeats.

In our case with the tipping percentages, we identify the mode by searching for the number that appears most often. Here, 27.9 is the only tip percentage repeating, occurring twice. Thus, our data set is unimodal with a mode of 27.9. This demonstrates that the most common tipping percentage among the sampled tables was 27.9%.

Standard Deviation Calculation

Standard deviation is a measure of the spread or dispersion of a set of values, indicating how much the individual values in a data set deviate from the mean on average. A lower standard deviation implies that the values tend to be close to the mean, while a higher standard deviation indicates that the values are spread out over a wider range.

To calculate the standard deviation, first, we determine the mean, then work out the difference between each data point and the mean, squaring the result. Averaging these squared differences gives us the variance, and finally, taking the square root of the variance gives us the standard deviation. For our tip percentage example, the calculated standard deviation is approximately 18.296, which suggests there is a considerable variation in the tipping percentages among the different tables.