Question

A geneticist counted the number of bristles on a certain region of the abdomen of the fruitfly Drosophila melanogaster. The results for 119 individuals were as shown in the table. \(^{60}\). $$ \begin{array}{|cccc|} \hline \begin{array}{c} \text { Number } \\ \text { of bristles } \end{array} & \begin{array}{c} \text { Number of } \\ \text { flies } \end{array} & \begin{array}{c} \text { Number of } \\ \text { bristles } \end{array} & \begin{array}{c} \text { Number } \\ \text { of flies } \end{array} \\ \hline 29 & 1 & 38 & 18 \\ 30 & 0 & 39 & 13 \\ 31 & 1 & 40 & 10 \\ 32 & 2 & 41 & 15 \\ 33 & 2 & 42 & 10 \\ 34 & 6 & 43 & 2 \\ 35 & 9 & 44 & 2 \\ 36 & 11 & 45 & 3 \\ 37 & 12 & 46 & 2 \\ \hline \end{array} $$ (a) Find the mean number of bristles. (b) Find the SD of the sample. (c) What percentage of the observations fall within 3 SDs of the mean? (d) What is the coefficient of variation?

Short answer

To find the mean, multiply the number of bristles by the corresponding number of flies, sum the results, and divide by the total number of flies. Calculate the SD using the sum of squared differences divided by one fewer than the total number of observations. The percentage of observations within 3 SDs of the mean is found by counting and comparing to the total. The coefficient of variation is the SD divided by the mean, expressed as a percentage.

Step-by-step solution

Finding the Mean

To find the mean number of bristles, multiply the number of bristles by the number of flies for each row to get the product. Sum all the products and then divide by the total number of flies (119). Mean = (Total sum of products)/(Total number of flies).

Calculating the Sum of Products

Add the products of number of bristles and number of flies for all the rows to get the total sum of products. For instance, for 29 bristles and 1 fly, the product is 29*1 = 29.

Calculating the Mean

Use the total sum obtained in Step 2 and divide it by 119 (the total number of flies) to get the mean.

Calculating the Variance

To find the variance, subtract the mean from each bristle count, square the results, multiply each by the respective number of flies, and sum those products. Finally, divide by the number of observations minus one (N-1, where N is the total number of flies). Variance = (Sum of squared differences)/(N-1).

Finding the Standard Deviation

Take the square root of the variance calculated in Step 4 to obtain the Standard Deviation (SD).

Calculating the Proportion Within 3 SDs

Count the number of observations (flies) that have bristle counts within 3 standard deviations of the mean. Divide this count by the total number of observations and then multiply by 100 to get the percentage.

Finding the Coefficient of Variation

Divide the standard deviation by the mean and then multiply by 100 to express it as a percentage. Coefficient of Variation = (SD/Mean) * 100.

Key concepts

Understanding Bristle Count in Drosophila melanogaster

In the study of genetics, scientists often count specific features of organisms, like the number of bristles on fruit flies. This count is crucial because it's a variable that can show variation due to genetic differences. As we see in our example, the bristle count is recorded for Drosophila melanogaster, which can further be used to find patterns and make inferences about the population.

For students needing to understand this concept, the bristle count is simply the number of hair-like appendages in a specified area. These counts become data points for statistical analysis. It's important to compile this data accurately, as it forms the basis for all subsequent calculations such as mean, variance, and standard deviation.

The Role of Standard Deviation

The standard deviation (SD) is a measure of the amount of variation or dispersion of a set of values. A low SD indicates that the values tend to be close to the mean of the set, while a high SD indicates that the values are spread out over a wider range.

In our fruit fly example, calculating the SD of bristle counts can show us how much variation there is in bristle number among the flies. To put it in perspective, if all flies had identical or very close bristle counts, the SD would be very low. However, a variety of bristle counts would result in a higher SD. This measure helps geneticists understand the diversity present in the population.

Coefficient of Variation: Comparing Variability

The coefficient of variation (CV) is another important statistical measure. It's a standardized measure of dispersion of a probability distribution or frequency distribution. It's expressed as a percentage and is calculated by dividing the SD by the mean and then multiplying by 100.

What makes the CV incredibly useful is its ability to compare variability between datasets that have different units or means. For example, if a researcher wanted to compare bristle count variability not just among Drosophila melanogaster, but across different species, the CV would be an appropriate measure to use. It allows a comparison of the degree of variation from one data series to another, even if the means are drastically different.

Variance Calculation and Its Importance

Variance is the average of the squared differences from the Mean. It's a figure that represents how much the bristle count for each fly deviates from the average bristle count.

To find the variance, one would subtract the mean from each individual bristle count, square that number to make it positive, multiply by the frequency of that bristle count, and then take an average of those values. Remember, when calculating sample variance, we divide by N-1 (where N is the number of observations) not just N. This is to correct the bias in the estimation of the population variance from a sample.

Why Calculate Variance?

Understanding variance helps in appreciating variability in the data. It's the foundation for many other statistical measures, such as standard deviation. When students grasp the concept of variance, they are better equipped to understand the spread and dispersion of data, which is pivotal for any statistical analysis.