Question

As part of a classic experiment on mutations, 10 aliquots of identical size were taken from the same culture of the bacterium \(E\). coli. For each aliquot, the number of bacteria resistant to a certain virus was determined. The results were as follows: \(^{24}\) $$\begin{array}{lllll}14 & 15 & 13 & 21 & 15 \\\14 & 26 & 16 & 20 & 13\end{array}$$ (a) Construct a frequency distribution of these data and display it as a histogram. (b) Determine the mean and the median of the data and mark their locations on the histogram.

Short answer

The histogram has classes plotted on the x-axis with corresponding frequency on the y-axis. In this dataset, the mean is 16.7 and the median is 15. Both values should be clearly marked on the histogram.

Step-by-step solution

Order the Data

First, arrange the data in ascending order to make the construction of the frequency distribution easier. The ordered data: 13, 13, 14, 14, 15, 15, 16, 20, 21, 26.

Determine Frequency Distribution

Create a table with two columns: one for the number of bacteria resistant to the virus (the 'class'), and one showing the frequency of each class. Count the number of occurrences for each class:| Number of Bacteria | Frequency ||--------|-----------|| 13 | 2 || 14 | 2 || 15 | 2 || 16 | 1 || 20 | 1 || 21 | 1 || 26 | 1 |

Draw the Histogram

On the x-axis, write the number of bacteria (class). On the y-axis, write the frequency. For each class, draw a bar that reaches up to the corresponding frequency.

Calculate Mean

To calculate the mean, sum all the observed values and divide by the total number of observations: \( \frac{13 + 13 + 14 + 14 + 15 + 15 + 16 + 20 + 21 + 26}{10} = \frac{167}{10} = 16.7 \).

Determine Median

As there are an even number of data points, the median will be the average of the two middle values. Since the data is arranged in ascending order, take the 5th and 6th values and find their average: \( \frac{15 + 15}{2} = \frac{30}{2} = 15 \) is the median.

Mark Mean and Median on Histogram

On the histogram, mark the mean and median. The mean is marked at 16.7 and the median at 15, which might be indicated using different colors or symbols on the histogram.

Key concepts

Histogram

Visualizing data is a critical step in understanding its distribution and structure, and a histogram is a powerful tool for such visualization. A histogram is a type of bar graph that displays the frequency of data within certain ranges, known as bins or classes. Each bin represents an interval of values, and the height of the bar correlates to the frequency of data points within that interval.

For example, consider a researcher conducting a bacteria resistance experiment who wants to graph the number of occurrences of resistant bacteria in several samples. They would start by sorting the number of bacteria into intervals. Then, they would plot these on the x-axis of a graph. On the y-axis, they would represent the frequency of each interval. By doing so, the resulting chart shows at a glance which ranges of bacteria counts are most common, allowing the researcher to quickly identify patterns or anomalies in the data.

Moreover, histograms are convenient for comparing the mean and median visually, providing insights into the data's skewness. A histogram where the mean and median are marked helps better understand the central tendency of the data, which can be essential when analyzing results from an experiment, like the spread of resistance among bacterial cultures.

Mean and Median Calculations

Mean and median are measures of central tendency that summarize a set of numbers using a single value, aiming to give an idea of the 'center' of the data set. The mean, often referred to as the average, is calculated by summing all the values and then dividing by the number of values. For instance, in a bacteria resistance experiment, adding together the total number of resistant bacteria across all samples and then dividing by the number of samples will give the mean value.

The median, on the other hand, is the middle number when the values are sorted in ascending order. If the data set has an odd number of observations, the median is the middle one. If there are an even number of observations, it is the average of the two central numbers. The median can be more informative than the mean in cases where the data includes outliers that may skew the mean. For example, if a single sample in a bacteria resistance experiment has an unusually high number of resistant bacteria, it may raise the mean, making the median a more representative measure for the rest of the samples.

Knowing where the mean and median fall in relation to each other can also provide insights into the data's distribution. If the histogram shows the mean and median at the same point, the distribution is likely symmetrical. If they differ, the data may be skewed to the left or right.

Bacteria Resistance Experiment

A bacteria resistance experiment is a scientific study exploring how bacterial populations withstand certain environmental pressures, such as antibiotic exposure or, in our example, a virus. By exposing the bacteria to these agents and recording the number of individuals that survive or resist, researchers can gain valuable insights into the rate of resistance, factors influencing resistance, and potential ways to overcome or mitigate it.

E. coli, a common bacterium used in laboratory experiments, serves as an excellent model for such studies due to its relatively simple genetics and quick reproduction rate. In a standard experiment, various aliquots or samples from a culture are exposed to the agent of interest. Scientists then observe and record the number of resistant bacteria in each sample. From this data, they can calculate the mean and median, construct histograms, and gain a better understanding of bacteria resistance within the population.

Such experiments are foundational in biology and medicine as they can directly inform healthcare practices and antibiotic development. Understanding how and when resistance develops enables researchers to predict bacterial behavior and potentially stave off the consequences of widespread resistance.