4.6.8 Mean & Standard Deviation

Key Terms

1. Mean:

• A measure of central tendency.
• Represents the average value of a dataset.

2. Standard Deviation (SD):

• A measure of spread or dispersion around the mean.
• Indicates how much the data varies from the mean value.

Key Formulae

1. $Mean (\bar{x}):$

$$\bar{x} = \frac{\Sigma x}{n}$$

Where:

• $\Sigma x$ = sum of all values in the dataset.
• $n$= number of values in the dataset.

2. Standard Deviation ($SD$):

$$SD = \sqrt{\frac{\Sigma (x - \bar{x})^2}{n - 1}}$$

Where:

• $x$ = each individual value.
• $\bar{x}$ = mean of the dataset.
• $n$= number of values in the dataset.

What They Tell Us

1. Mean:

• A higher or lower mean can indicate trends between populations or environmental effects.

2. Standard Deviation:

• A small SD means most data points are close to the mean (low variation).
• A large SD means data points are spread out from the mean (high variation).

Application in Biology

1. Comparing Populations:

• Use the mean to compare average characteristics (e.g., height, leaf size) between populations.

2. Evaluating Variation:

• Use SD to assess genetic or environmental variation within a population.
• Overlapping SD values between two datasets indicate no significant difference.

Worked Example

Scenario: Two populations of plants are studied for leaf length.

Data for Population A:

• Mean leaf length = 6 cm

• SD = 1 cm Data for Population B:

• Mean leaf length = 10 cm

• SD = 0.5 cm Interpretation:

• Population B has longer leaves on average (higher mean).

• Population A shows more variation in leaf length (larger SD).