Question

The weight gains of beef steers were measured over a 140 -day test period. The average daily gains (lb/day) of 9 steers on the same diet were as follows: \(^{23}\) $$\begin{array}{lllll}3.89 & 3.51 & 3.97 & 3.31 & 3.21 \\\3.36 & 3.67 & 3.24 & 3.27 &\end{array}$$ Determine the mean and median.

Short answer

The mean weight gain is 3.4922 lb/day, and the median weight gain is 3.36 lb/day.

Step-by-step solution

List the Data

Firstly, write down all the weight gains in ascending order. The gains are: 3.21, 3.24, 3.27, 3.31, 3.36, 3.51, 3.67, 3.89, 3.97.

Calculate the Mean

Add all the weight gains together and divide by the number of steers to get the mean weight gain. Mean = \((3.21 + 3.24 + 3.27 + 3.31 + 3.36 + 3.51 + 3.67 + 3.89 + 3.97) / 9\).

Calculate the Mean (cont.)

Calculate the sum: \(3.21 + 3.24 + 3.27 + 3.31 + 3.36 + 3.51 + 3.67 + 3.89 + 3.97 = 31.43\). Now divide the sum by 9: \(31.43 / 9 = 3.4922\) lb/day.

Find the Median

The median is the middle value when the data is ordered. Since there are 9 data points, the median will be the 5th value. Therefore, the median is 3.36 lb/day.

Key concepts

Mean Calculation

Understanding the mean calculation is essential in statistics as it represents the average value in a data set. To compute it, you sum all the values and divide this sum by the number of values present. For example, in the case of the beef steers' weight gains over 140 days, you would add all daily gains together and then divide by the number of steers, which is 9 in this instance. This calculation results in the mean weight gain.
The simplicity of mean calculation makes it a popular statistical measure, but it's worth noting that it can be affected by extreme values, known as outliers, which can skew the result. Nonetheless, if all values are typical and there are no significant outliers, the mean provides an excellent overview of the central tendency in a dataset.

Median Calculation

In contrast to the mean, the median calculation identifies the middle value in an ordered dataset. It is found by arranging the data points from smallest to largest and selecting the one right in the center. Should the dataset contain an even number of observations, the median is the average of the two middle numbers.
For the steers' weight gains, after ordering the gains, the 5th value in the sequence becomes the median since there are an odd number of data points. The significance of the median lies in its resistance to the influence of outliers; it provides a more accurate central point in datasets with skewed data or outliers.

Data Analysis in Life Sciences

The application of data analysis in life sciences spans a wide array of fields, from ecology to genetics, and is crucial for interpreting experimental results and making evidence-based decisions. It involves collecting, processing, and analyzing biological data to understand various phenomena.
In biological data analysis, statisticians often grapple with issues like variable sample sizes, noisy data, and complex interactions. Measures such as the mean and median can help summarize and clarify these data. Importantly, researchers must ensure that the data analysis methods chosen are appropriate for the nature of their data and the questions they are examining.

Statistical Measures of Central Tendency

The statistical measures of central tendency give us an indication of the average or most typical value of a dataset. The most commonly used measures are the mean, median, and mode. Each offers insights into the dataset's characteristics and helps summarize large amounts of data in a single value.
Choosing the right measure of central tendency depends on the dataset's structure and the presence of outliers. While the mean offers a quick arithmetic average, the median can provide a midpoint that is not skewed by outliers, and the mode reflects the most frequently occurring value in the dataset. These measures are fundamental in statistical analysis and are widely used in the life sciences to interpret experiment and observation data.