Problem 7 - Algebra Form and Function

Question

The clutch size of a bird is the number of eggs laid by the bird. Table 17.7 shows the clutch size of six different birds labeled (i)-(vi). What is the (a) Mean clutch size? (b) Standard deviation of these clutch sizes? $$ \begin{array}{c|c|c|c|c|c|c} \hline \text { Bird } & \text { (i) } & \text { (ii) } & \text { (iii) } & \text { (iv) } & \text { (v) } & \text { (vi) } \\ \hline \text { Clutch size } & 6 & 7 & 2 & 3 & 7 & 5 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Answer: The mean clutch size is 5, and the standard deviation is approximately 2.097.

Step 1: Identify Data

Clutch sizes for the six birds are: 6, 7, 2, 3, 7, and 5.

Step 2: Calculate the total sum of clutch sizes

Sum the clutch sizes: 6 + 7 + 2 + 3 + 7 + 5 = 30

Step 3: Calculate the Mean

Divide the total sum of clutch sizes by the number of birds (6) to find the mean clutch size: $$\frac {30}{6} = 5$$ Mean clutch size = 5 #b# Standard Deviation of Clutch Sizes:

Step 1: Identify Data

We will use the same clutch size data as in part a.

Step 2: Calculate the mean

Use the mean calculated in part a, which is 5.

Step 3: Calculate deviation from the mean for each clutch size

Subtract the mean from each clutch size and make a list of the differences: Bird (i): 6 - 5 = 1 Bird (ii): 7 - 5 = 2 Bird (iii): 2 - 5 = -3 Bird (iv): 3 - 5 = -2 Bird (v): 7 - 5 = 2 Bird (vi): 5 - 5 = 0

Step 4: Square the deviations

Square each of the differences obtained in step 3: Bird (i): 1^2 = 1 Bird (ii): 2^2 = 4 Bird (iii): (-3)^2 = 9 Bird (iv): (-2)^2 = 4 Bird (v): 2^2 = 4 Bird (vi): 0^2 = 0

Step 5: Calculate the sum of the squared deviations

Add all the squared deviations obtained in step 4: 1 + 4 + 9 + 4 + 4 + 0 = 22

Step 6: Calculate the variance

Divide the sum of squared deviations by the number of birds (6) minus 1: $$\frac{22}{6-1} = \frac{22}{5} = 4.4$$ Variance = 4.4

Step 7: Calculate the standard deviation

Calculate the square root of the variance to find the standard deviation: $$\sqrt{4.4} \approx 2.097$$ Standard deviation of clutch sizes is approximately 2.097.

Key Concepts

Understanding Clutch Size

Clutch size refers to the number of eggs that a bird lays in one nesting period. It provides vital information about the reproductive strategy of the bird species. Understanding the clutch size is useful in ecological and biological studies to assess how birds adapt to their environments. This can relate to

• Breeding success, where larger clutch sizes can lead to more offspring.
• Environmental pressures, since changes in available resources may influence how many eggs a bird can successfully hatch and rear.
• Conservation efforts, as shifts in clutch sizes can indicate changes in habitat quality or the effects of climate change.

With data gathered on clutch sizes, researchers can perform statistical analysis to determine patterns or variances in reproductive success or adaptational strategies across different bird species.

Overview of Statistical Analysis

Statistical analysis involves interpreting data to derive meaningful insights. In the context of clutch sizes, statistical methods help estimate the average clutch size (mean) and the variation around that average (standard deviation). This information is important because it indicates:

• The central tendency, or what the typical clutch size is among a group of birds.
• The spread of the data, which tells us whether most birds are laying a similar number of eggs or if there is wide variation in the group.

Statistical analysis helps in making informed decisions based on the data collected. It allows researchers to assess differences and identify patterns that may warrant further investigation. Through these analyses, scientists can understand whether the clutch sizes observed are typical or if anomalies or trends are occurring.

Examining Variance in Data

Variance measures how far a set of numbers are spread out from their mean. In the study of clutch sizes, variance gives us an idea of the diversity of clutch sizes within a group of birds. A low variance indicates that the numbers are close to the mean, while a high variance suggests greater disparity.

The formula for variance involves squaring the deviations of each data point from the mean, summing these squared values, and then dividing by the number of data points minus one. It can be useful to compare variances across different species or populations to identify ecological or evolutionary implications.

• Low variance might suggest a stable environment where optimal clutch size is more consistent.
• High variance could indicate varied conditions or selective pressures affecting reproductive strategies.

Interpreting variance alongside the mean and standard deviation allows a deeper understanding of the reproductive patterns of birds and how these might shift in response to environmental changes.