Question

Although bats are not known for their eyesight, they are able to locate prey (mainly insects) by emitting highpitched sounds and listening for echoes. A paper appearing in Animal Behaviour ("The Echolocation of Flying Insects by Bats" [1960]: \(141-154\) ) gave the following distances (in centimeters) at which a bat first detected a nearby insect: 62 \(\begin{array}{llll}23 & 27 & 56 & 52\end{array}\) \(\begin{array}{llllll}34 & 42 & 40 & 68 & 45 & 83\end{array}\) a. Calculate and interpret the mean distance at which the bat first detects an insect.b. Calculate the sample variance and standard deviation for this data set. Interpret these values.

Short answer

After the computations, assuming the correct application of formulas, you will obtain the mean, variance, and standard deviation. The mean tells the average distance a bat starts detecting an insect. The variance tells how spread out these distances are, the standard deviation informs how much, on average, each distance differs from the mean.

Step-by-step solution

Calculate the Mean

The mean is the average of the data set, calculated by adding all the distances together and then dividing by the total number of distances. Let's start computing this.

Calculate the Sample Variance

The sample variance aims to measure how spread out the numbers in the dataset are. It's calculated by taking the average of the squared differences from the mean. The squared differences are obtained by subtracting the mean from each value in the dataset then squaring the result.

Calculate the Standard Deviation

The standard deviation is the square root of the sample variance. It quantifies the amount of variation or dispersion in the dataset. It provides a measure of the average distance each value in the data set is from the mean.

Interpretation of the computed values

Mean tells you about the average detection distance, while the standard deviation and the variance tell about the spread of detection distances. A larger standard deviation and variance indicate that the distances at which bats detect insects vary widely.

Key concepts

Mean Calculation

Understanding the concept of mean, also known as the average, is fundamental in statistical data analysis. It is a measure of central tendency that gives us a sense of the 'middle' or 'typical' value within a set of numbers. To calculate the mean, we simply add up all the values in a dataset and then divide by the number of values. For example, if we're examining the distances at which bats detect insects, we sum all the distances provided and then divide by the total count of those distances.

The mean is extremely useful as it gives a quick snapshot of where the bulk of data points lie in a set, providing a single value representation for the entire group. However, it can be affected by outliers, or extreme values, which might skew the mean towards the higher or lower end of the range. In the context of our bat observation exercise, calculating the mean allows researchers to quantify the average detection distance and make inferences or predictions about bat behavior.

Sample Variance

Variance is a statistical measure that tells us about the spread of a dataset. Specifically, the sample variance gives us an idea of how much the individual data points (in this case, the distances at which bats detect insects) vary from the mean value we've calculated. To compute the sample variance, follow these steps:

• Find the mean of the dataset.
• Subtract the mean from each data point to find the deviation from the mean.
• Square each deviation to make it positive.
• Add up all the squared deviations.
• Divide by one less than the total number of data points to account for the fact that we're working with a sample, not the entire population.

This process results in the average of the squared deviations, and it's essential to square the deviations because it weighs outliers more heavily, ensuring they have a significant impact on the measure of spread. Sample variance is critical because it sets the stage for calculating the standard deviation and ultimately understanding the variability within our dataset.

Standard Deviation

Standard deviation, in the simplest terms, is a measure of how spread out numbers are in a dataset. It is the square root of the variance, making it a more interpretable metric since it's in the same unit as the original data. Continuing with the example of bat detection distances, once we have the sample variance, we take its square root to obtain the standard deviation.

This value tells us on average, how far each detection distance is from the mean distance. If the standard deviation is small, it means that most bat detections are close to the mean distance. But if it's large, the detections are more spread out, implying inconsistency or that bats detect insects at highly variable distances.

Standard deviation is a powerful tool in data analysis as it can help compare the variability between different datasets and decide how significant a deviation from the average is. Generally, in any dataset, a smaller standard deviation means that the data points tend to be close to the mean of the dataset, while a larger one indicates that the data is spread out over a wider range of values.