Question

In Example 4.1 .2 it was stated that bill length in a population of Blue Jays follow a normal distribution with mean \(\mu=25.4 \mathrm{~mm}\) and standard deviation \(\sigma=0.8 \mathrm{~mm} .\) Use the \(68 \%-95 \%-99.7 \%\) rule to determine intervals, centered at the mean, that include \(68 \%, 95 \%,\) and \(99.7 \%\) of the bill length in the distribution.

Short answer

The intervals centered at the mean are 24.6 mm to 26.2 mm for 68%, 23.8 mm to 27.0 mm for 95%, and 23.0 mm to 27.8 mm for 99.7% of the bill lengths.

Step-by-step solution

Understanding the 68-95-99.7% Rule

The 68-95-99.7% rule, also known as the empirical rule, tells us that for a normal distribution: approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. Our aim is to find these intervals centered around the mean.

Calculate the Interval for 68%

To find the interval that includes 68% of bill lengths, determine one standard deviation above and below the mean. Calculate the lower end of the interval by subtracting one standard deviation from the mean: 25.4 mm - 0.8 mm = 24.6 mm. Calculate the upper end of the interval by adding one standard deviation to the mean: 25.4 mm + 0.8 mm = 26.2 mm.

Calculate the Interval for 95%

To find the interval that includes 95% of bill lengths, determine two standard deviations above and below the mean. Calculate the lower end of the interval by subtracting two standard deviations from the mean: 25.4 mm - 2(0.8 mm) = 23.8 mm. Calculate the upper end of the interval by adding two standard deviations to the mean: 25.4 mm + 2(0.8 mm) = 27.0 mm.

Calculate the Interval for 99.7%

To find the interval that includes 99.7% of bill lengths, determine three standard deviations above and below the mean. Calculate the lower end of the interval by subtracting three standard deviations from the mean: 25.4 mm - 3(0.8 mm) = 23.0 mm. Calculate the upper end of the interval by adding three standard deviations to the mean: 25.4 mm + 3(0.8 mm) = 27.8 mm.

Key concepts

Normal Distribution

Imagine you're observing various characteristics in nature, like the height of trees or the bill length of Blue Jays. You'll often notice that most of the observations tend to cluster around a central value with fewer and fewer occurrences as you move away from it. This pattern of data is what we call a 'normal distribution', a key concept in statistics often visualized as a symmetrical, bell-shaped curve.

When a dataset is normally distributed, it follows certain properties: the mean (average), median (middle value), and mode (most frequent value) all coincide at the center of the distribution. It's as if nature loves balance; most things are average, and extremes are rare. This distribution is incredibly common in the natural and social sciences for variables that come under the influence of many small, random disturbances.

Now, in the context of our feathered friends, the Blue Jays, if we measure the bill lengths of a large population, we'll find that most bills are around the average length, with only a few notably shorter or longer, creating that classic bell curve shape reflective of a normal distribution.

Standard Deviation

Now, let's talk about 'variability' or how spread out individual measurements are. In the quest for understanding how much variation exists, we use a measurement called 'standard deviation'. Think of it as a ruler that measures how far a particular value is from the 'norm', giving us a sense of the average distance between the values in a dataset and the mean.

Using the bill length example, a standard deviation of 0.8 mm doesn't just give us a number; it implies that the bill lengths of Blue Jays don't vary wildly. They're fairly consistent, only deviating from the average by a small amount on average. If the standard deviation were larger, it would mean that there is more diversity in bill size within the population.

Importance of Standard Deviation

• Understanding Variability: It tells us how much variation or 'dispersion' there is from the average (mean).
• Comparing Datasets: It allows scientists to compare the dispersion of traits across different populations.
• Identifying Extremes: Standard deviation helps in identifying the 'unusual' (extremely large or small) observations.

Empirical Rule

The empirical rule, famously captured by the 68-95-99.7% rule, is like a lighthouse guiding ships safely through the foggy waters of statistics. It allows us to make quick, practical estimates about the distribution of data within a normal distribution.

According to the rule, approximations can be made such as: 68% of the data falls within one standard deviation of the mean; 95% within two standard deviations; and an overwhelming 99.7% falls within three standard deviations. Why is this helpful? Well, with this rule, without doing any complex calculations, we can estimate where most of our data points fall and how rare certain measurements are.

For instance, knowing that 95% of the bill lengths are within 23.8 mm and 27.0 mm gives direct insight for ecologists studying Blue Jays. It sets a predictable range for 'typical' bill lengths and highlights the 'atypical' ones, offering clues to genetic and environmental factors at play. In essence, the empirical rule enables us to harness the predictive power of the normal distribution, illuminating patterns and variances within complex data.