A histogram is the graphical representation of a continuous frequency distribution. Unlike a bar chart, there are no gaps between the bars because the underlying variable is continuous (e.g., body mass of lizards, fish length).
- X-axis: Class intervals (continuous boundaries).
- Y-axis: Frequency or Frequency Density.
Key Concept (Frequency Density):
If class intervals are unequal, the height of the bar is not the frequency, but the frequency density, calculated as:
Frequency Density = Class Frequency / Class Width
CSIR/Exam Note: The area of each bar is proportional to the frequency of that class. The total area of the histogram represents the total number of observations (N).
A frequency polygon is a line graph that outlines the shape of the data distribution. It is particularly useful for comparing two or more statistical distributions on the same axes.
Construction: It is derived by plotting the class marks (midpoints) of the class intervals on the X-axis against their corresponding frequencies on the Y-axis. The points are then joined by straight lines.
Closing the Polygon: To make it a closed polygon (touching the X-axis), we imagine a pre-first class and a post-last class with zero frequency and extend the lines to their midpoints.
Mathematical Property: The total area enclosed by a frequency polygon is precisely equal to the total area of the corresponding histogram.
When the sample size (N) approaches infinity and the class intervals become infinitely small, the jagged lines of a frequency polygon smooth out into a continuous, flowing curve called a Frequency Curve.
In biological systems, these curves generally take four major shapes:
Symmetrical (Normal/Bell-shaped)
Mean = Median = Mode. (e.g., human birth weight).
Positively Skewed (Right-skewed)
Tail extends to the right. Mean > Median > Mode. (e.g., distribution of a rare parasite among a host population).
Negatively Skewed (Left-skewed)
Tail extends to the left. Mean < Median < Mode. (e.g., age at death in a stable, well-nourished population).
J-shaped / U-shaped
Extreme distributions where frequencies peak at one or both ends.
Sometimes, absolute frequencies are misleading, especially when comparing two populations of different sample sizes (e.g., comparing 50 frogs from Pond A with 500 frogs from Pond B).
Definition: Relative frequency is the fraction or percentage of the total number of observations falling into a specific class.
Relative Frequency = f / N
(where f is the class frequency and N is the total frequency)
Properties:
- The sum of all relative frequencies always equals 1.0 (or 100%).
- It effectively transforms a frequency distribution into a probability distribution, bridging descriptive statistics and inferential statistics.
Advanced Scenario:
Imagine you are analyzing the shell thickness of a gastropod population. The data yields a highly positively skewed frequency curve.
If you were to randomly pick a gastropod from this population, would its shell thickness most likely be closer to the statistical Mean or the Mode of the dataset?
Hint: Think about where the "peak" of the curve lies compared to the long tail.