Problem 10 - Algebra Form and Function

Question

A naturalist collects samples of a species of lizard and measures their lengths. Give the (a) sample size (b) mean (c) range (d) \(\quad\) standard deviation. $$ \begin{array}{l|c|c|c|c|c} \hline \text { Lizard no. } & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Length }(\mathrm{cm}) & 5.8 & 5.9 & 5.9 & 6.0 & 6.5 \\ \hline \text { Lizard no. } & 6 & 7 & 8 & 9 & 10 \\ \hline \text { Length }(\mathrm{cm}) & 7.9 & 7.9 & 8.0 & 8.0 & 8.1 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Answer: The sample size is 10, the mean is 7.0 cm, the range is 2.3 cm, and the standard deviation is approximately 1.0 cm.

Step 1: Sample Size

Count the number of lizards in the dataset. The sample size is 10.

Step 2: Mean

To find the mean, sum the lengths of all the lizards and divide by the sample size. $$ \text{Mean} = \frac{\text{Sum of Lengths}}{\text{Sample Size}} = \frac{5.8+5.9+5.9+6.0+6.5+7.9+7.9+8.0+8.0+8.1}{10} = \frac{70.0}{10} = 7.0 \thinspace \mathrm{cm} $$

Step 3: Range

To find the range, subtract the minimum length from the maximum length. $$ \text{Range} = \text{Maximum Length} - \text{Minimum Length} = 8.1 - 5.8 = 2.3 \thinspace \mathrm{cm} $$

Step 4: Standard Deviation

To find the standard deviation, compute the square root of the average of the squared differences from the mean. $$ \text{Standard Deviation} = \sqrt{\frac{\sum_{i=1}^{10}(x_i - \text{Mean})^2}{10}} $$ Compute the squared differences from the mean for each length, sum them up, and divide by the sample size: $$ \begin{aligned} \frac{(5.8-7.0)^2 +(5.9-7.0)^2 +(5.9-7.0)^2+&\dots +(8.0-7.0)^2 +(8.1-7.0)^2}{10} \\ =\frac{1.44+1.21+1.21+1.0+0.25+0.81+0.81+1.0+1.0+1.21}{10} \\ =\frac{9.94}{10} \end{aligned} $$ Take the square root of the result: $$ \text{Standard Deviation} = \sqrt{\frac{9.94}{10}} \approx 1.0 \thinspace \mathrm{cm} $$ So, the results are: (a) Sample size: 10 (b) Mean: 7.0 cm (c) Range: 2.3 cm (d) Standard Deviation: 1.0 cm

Key Concepts

Sample Size

The sample size is an important concept in statistics that tells us how many individual observations or measurements are in a dataset. It represents the total number of samples collected. In our lizard example, the sample size helps us understand the total count of lizards that were measured. To determine the sample size, simply count each lizard measured in the dataset.

• For the lizard length data provided, the sample size is 10.

A larger sample size generally increases the reliability of the statistical analysis, as it encompasses more data variation, giving us a better view of the whole population from which these samples were drawn.

Mean Calculation

The mean, often referred to as the average, is a measure that summarizes a set of numbers by focusing on their central tendency. Calculating the mean provides a single value that represents the typical value of a dataset. Here's how to calculate the mean:

• Add up all the lengths of the lizards.
• Divide the sum by the total number of lizards (which is the sample size).

In our example, the mean is calculated by taking the total sum of the lizard lengths, which is 70.0 cm, and dividing it by the sample size of 10.
This gives us a mean length of 7.0 cm, meaning that, on average, lizards in this sample are 7.0 cm long.

Range in Data

The range in data gives us a basic measure of variability by indicating the difference between the largest and smallest values in a dataset. It represents the spread of the data and highlights how dispersed or clustered the values are. To find the range:

• Identify the maximum and minimum values in the dataset.
• Subtract the smallest value from the largest value.

In the case of the lizard length data, the largest length is 8.1 cm and the smallest is 5.8 cm. Thus, the range is calculated as 8.1 cm - 5.8 cm = 2.3 cm.
This range tells us about the span within which all lizard lengths lie, shedding light on how varied the sizes are.

Standard Deviation Calculation

Standard deviation is a crucial statistic that measures the amount of variation or dispersion in a set of values. It tells us how much the individual data points typically vary from the mean, providing insight into the spread of data. Here's how to calculate it:

• Subtract the mean from each data point and square the result.
• Sum all squared differences.
• Divide the sum by the number of samples.
• Take the square root of this quotient.

For the lizard dataset, each lizard's length is subtracted from the mean length (7.0 cm), squared, and then these squared differences are summed up. The result is divided by the number of lizards, and the square root is taken, yielding a standard deviation of approximately 1.0 cm.
This statistic helps us understand that the lengths of lizards vary about 1.0 cm from the mean length in our sample.