Histograms - Algebra-1

1. Fundamental Concepts

Definition: A histogram is a statistical graph used to display the distribution of continuous or numerical data. It consists of a series of rectangles (bars) with equal width, where each rectangle represents the frequency (or count) of data within a specific interval.
Purpose: It visually illustrates the distribution pattern of data, such as central tendency, dispersion, and symmetry, making it suitable for large datasets or those with a wide range of values.
Difference from other graphs: Unlike bar charts (used for discrete categories with gaps between bars), histograms have no gaps between rectangles (to reflect data continuity). The width of each rectangle represents the interval range, and the height represents the frequency of that interval.

2. Key Concepts

Intervals/Bins: The core element of a histogram. Data is divided into continuous intervals of equal width, with no overlaps or gaps between intervals to ensure full coverage of the dataset.
Interval Width: The range of values in each interval (e.g., 1–5 has a width of 4). Consistent width is mandatory; otherwise, the visual representation of data distribution will be distorted.
Frequency: The number of data points falling within each interval, corresponding to the height of the rectangle in the histogram. A taller rectangle indicates a higher concentration of data in that interval.
Axes: The x-axis (horizontal axis) represents the interval ranges, while the y-axis (vertical axis) represents the frequency (or count) of data in each interval.

3. Examples

Easy
Dataset: {1, 1, 2, 3, 3, 3, 4, 4}
Intervals: [1–2], [3–4] (both with a width of 2)
Task: Draw the histogram and describe the distribution.
Answer:
- The interval [1–2] contains data points 1, 1, 2, with a frequency of 3, so the corresponding rectangle has a height of 3.
- The interval [3–4] contains data points 3, 3, 3, 4, 4, with a frequency of 5, so the corresponding rectangle has a height of 5.
- Distribution feature: Data is more concentrated in the [3–4] interval.
Medium
Dataset: {5, 7, 5, 9, 7, 7, 11, 9, 5, 13}
Intervals: [5–7], [8–10], [11–13] (each with a width of 3)
Task: (1) Calculate the frequency of each interval; (2) Draw the histogram and identify the interval with the highest frequency.
Answer:
- Interval [5–7] includes 5, 5, 5, 7, 7, 7, with a frequency of 6.
- Interval [8–10] includes 9, 9, with a frequency of 2.
- Interval [11–13] includes 11, 13, with a frequency of 2.
- The interval with the highest frequency is [5–7].
Hard
Dataset: {0, 0, 1, 1, 2, 3, 3, 3, 3, 3, 3, 9, 15}
Intervals: [0–2], [3–8], [9–16] (with appropriate equal width adapting to the data range)
Task: (1) Draw the histogram; (2) Analyze the distribution characteristics (e.g., whether data is concentrated in a specific interval).
Answer:
- Interval [0–2] has a frequency of 5 (includes 0, 0, 1, 1, 2).
- Interval [3–8] has a frequency of 6 (includes six 3s).
- Interval [9–16] has a frequency of 2 (includes 9, 15).
- Distribution feature: Data is concentrated in the [3–8] interval, with sparse data in the higher-value interval (right side), showing a right-skewed distribution.

4. Problem-Solving Techniques

Step 1: Determine the data range
Calculate the minimum and maximum values of the dataset to determine the total range (maximum - minimum), which guides interval division.
Step 2: Divide into equal-width intervals
Based on the data range and size, reasonably set the number of intervals (usually 5–10), ensuring each interval has equal width with no overlaps or gaps. For example, data ranging from 0 to 15 can be divided into [0–3], [4–7], etc. (width = 3).
Step 3: Count frequencies for each interval
Check each data point to count how many fall into each interval (frequency), which can be assisted by a frequency table.
Step 4: Draw the histogram
Mark intervals on the x-axis and frequencies on the y-axis. Draw rectangles with heights corresponding to the frequencies, leaving no gaps between rectangles.
Step 5: Analyze distribution characteristics
Observe the height changes of rectangles to identify the concentrated interval, symmetry (e.g., symmetric, left-skewed, right-skewed), and interpret information based on specific questions.