Variation.12Aug2025.Lecture8

​12Aug2025.Lecture8

Measures of Variation

By J Arro

Multiple Choice

Two lakes both have fish averaging 10 cm in length. In Lake A, all fish are exactly 10 cm. In Lake B, fish range from 6 to 14 cm. Which lake is likely to have a more complex food web?

• Lake A

• Lake B

• Both equally complex

• Cannot tell without knowing the mean

Multiple Choice

What is the range of fish lengths in Lake B (6, 8, 10, 12, 14 cm)?

• 4

• 6

• 8

• 10

Multiple Choice

IQR is the spread of the middle 50% of the data.

For the same fish lengths (6, 8, 10, 12, 14 cm), what is the Interquartile Range (IQR)?

• 4 cm

• 6 cm

• 8 cm

• 2 cm

Multiple Choice

In a boxplot comparing Lake A and Lake B, which lake’s plot would have a wider box and whiskers?

• Lake A

• Lake B

• Both the same

• Cannot determine

Multiple Choice

Which statement about Standard Deviation (SD) is true?

• High SD means values are closer to the mean.

• Low SD means data is more spread out.

• High SD means values are more spread out

• SD is unaffected by extreme values.

Multiple Choice

Why might a plant species with high variation in chemical defenses be more resilient?

• All plants taste the same to herbivores.

• Some plants deter herbivores while others don’t, supporting varied interactions

• Herbivores can eat everything equally.

• Variation confuses predators but harms the plants.

Multiple Choice

Which statistical fact explains why two datasets can have the same mean but different ecological outcomes?

• The datasets have different variances.

• The datasets have different medians.

• The datasets have different sums.

• The datasets have different modes.

Multiple Choice

A histogram with a long tail to the right is described as:

• Left-skewed

• Right-skewed

• Symmetrical

• Normal

Multiple Choice

In a fish population, most individuals are small but a few grow very large. This size distribution is:

• Symmetrical

• Right-skewed

• Left-skewed

• Bell-shaped

Multiple Choice

Why might fisheries management that uses only the average size be risky?

• It ignores variation that affects breeding capacity.

• It overestimates the number of large fish.

• It underestimates the smallest fish.

• It has no effect if the mean is correct.

Multiple Choice

Which best describes the role of variation in data analysis?

• Variation is unwanted noise.

• Variation contains important information.

• Variation should always be minimized.

• Variation means the data is invalid.

Word Cloud

In one word, why does variation matter?

Word Cloud

In one word, what is a mean?

Word Cloud

In one word, what is a standard deviation?

VARIATION part II

​14Aug2025

Multiple Choice

Bridge idea: Which pairing best builds intuition for the bell curve?

• Pie chart + scatter plot

• Histogram (raw) + density line (smooth) when there is variation

• Boxplot alone

• Bar plot of categories

​

Feature	Bar plot	Histogram

Data type (x-axis)

Categorical (factor/character) or precomputed summaries

Numeric (continuous) values

What height means

Count per category (geom_bar), or a supplied value/mean (geom_col)

Count or density in each bin (range)

Bar spacing

Gaps between categories

Touching bars (adjacent bins)

Area interpretation

Not meaningful (nominal scale)

Area ≈ proportion; density integrates to 1

Typical ggplot layer

geom_bar() (counts) or geom_col() (precomputed)

geom_histogram() (optionally aes(y = after_stat(density)))

Good add-on

Labels, reorder, percentages

Density curve (geom_density) for smoothing

==== Data Visualization: Histograms and Bar Plots ====

Multiple Choice

When all observed values are the same (e.g., all 155 cm), the sample SD is…

• Negative

• 0

• Undefined

• Always 1

Multiple Choice

The sample variance (s²) is best described as:

• The average of raw deviations from the mean

• The average of squared deviations from the mean (with a small-sample correction)

• The square of the mean

• The average absolute deviation

Multiple Choice

Which statement is true about variance and standard deviation?

• Both have the same units as the data

• Variance has squared units; SD is the square root of variance and has the original units

• Both are unitless

• SD has squared units; variance has the original units