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Consider the 13 girls mentioned in Exercise 2.6 .13 . At age 18 their average height was \(166.3 \mathrm{~cm}\) and the \(\mathrm{SD}\) of their heights was \(6.8 \mathrm{~cm} .\) Calculate the coefficient of variation.

Short Answer

Expert verified
The coefficient of variation for the heights is 4.09%.

Step by step solution

01

Understand the Coefficient of Variation

The coefficient of variation (CV) is a measure of relative variability. It is calculated as the ratio of the standard deviation (SD) to the mean, and is usually expressed as a percentage. The formula to calculate the CV is: CV = (SD / Mean) x 100%.
02

Insert the Mean and SD into the CV Formula

Insert the given mean height of 166.3 cm and the SD of 6.8 cm into the CV formula. CV = (6.8 / 166.3) x 100%.
03

Calculate the Coefficient of Variation

Perform the division and multiplication to calculate the coefficient of variation. CV = (0.0409) x 100% = 4.09%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Standard Deviation
Standard deviation (SD) is a cornerstone concept in the field of statistics, especially when assessing the spread or dispersion of a set of data points in relation to their mean (average). The SD quantifies the amount of variation or dispersion in a set of numeric values.

Why is standard deviation important? When you want to understand how much the individual data points, like the heights of girls in a certain group, deviate from the average height, you look at the standard deviation. A small SD indicates that the data points tend to be close to the mean, whereas a large SD shows that the data points are spread out over a wide range of values.

In the educational example offered, the SD of the girls' heights is 6.8 cm. This means that on average, the individual heights deviate from the mean height of 166.3 cm by 6.8 cm. Understanding this measure helps researchers in life sciences, for instance, to assess the genetic and environmental factors contributing to physical characteristics in a population.
Relative Variability and Its Role
Relative variability comes into play when we compare datasets with different units of measurement or vastly different mean values. In simple terms, it's like comparing apples to oranges. To make a fair comparison, we need a standardized measure that accounts for these differences – this is where the coefficient of variation (CV) is useful.

The CV expresses the standard deviation as a percentage of the mean, effectively normalizing the data. This makes it an excellent tool for comparison. For instance, if you're comparing the variability of heights in two different age-cohorts, using the CV will allow for a meaningful comparison regardless of the actual height differences.

This concept is especially useful in the life sciences, where biological variables can span a broad range of scales and a simple SD might be misleading. In the stated exercise, the calculation showed a CV of 4.09%, which informs us that the spread of the girls' heights is relatively small in comparison to their average height.
Statistics for Life Sciences
Statistics for life sciences is a vital field that enables scientists and researchers to organize, summarize, and interpret data related to living organisms and health-related issues.

Key statistical tools like mean, standard deviation, and the coefficient of variation offer insights into biological data sets that might otherwise seem puzzling. Whether one is studying growth patterns, like the height of girls at age 18, or more complex phenomena such as the spread of a disease, these statistical measurements provide clarity.

For future life scientists or those working in the field, mastering these concepts is essential. These tools help in determining the effectiveness of treatments, understanding genetic distribution, and much more. Life sciences heavily rely on accurate statistical analysis to make informed decisions and to contribute to the expanding collection of scientific knowledge.

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Most popular questions from this chapter

Trypanosomes are parasites that cause disease in humans and animals. In an early study of trypanosome morphology, researchers measured the lengths of 500 individual trypanosomes taken from the blood of a rat. The results are summarized in the accompanying frequency distribution. \({ }^{18}\) $$\begin{array}{|cccc|}\hline \begin{array}{c}\text { Length } \\\\(\mu \mathrm{m})\end{array} & \begin{array}{c}\text { Frequency } \\\\\text { (number of } \\\\\text { individuals) }\end{array} &\begin{array}{c}\text { Length } \\\\(\mu \mathrm{m})\end{array} & \begin{array}{c}\text { Frequency } \\\\\text { (number of } \\\\\text { individuals) }\end{array} \\\\\hline 15 & 1 & 27 & 36 \\\16 & 3&28 & 41 \\\17 & 21 & 29 & 48 \\\18 & 27 & 30 & 28 \\\19 & 23 & 31 & 43 \\\20 & 15 & 32 & 27 \\\21 & 10 & 33 & 23 \\\22 & 15 & 34 & 10 \\\23 & 19 & 35 & 4 \\\24 & 21 & 36 & 5 \\\25 & 34&37 & 1 \\\26 & 44 & 38 & 1 \\\\\hline\end{array}$$ (a) Construct a histogram of the data using 24 classes (i.e., one class for each integer length, from 15 to 38 ). (b) What feature of the histogram suggests the interpretation that the 500 individuals are a mixture of two distinct types? (c) Construct a histogram of the data using only 6 classes. Discuss how this histogram gives a qualitatively different impression than the histogram from part (a).

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