Histograms Explained

Histograms

• A histogram is a bar graph that organizes quantitative data.

Key Characteristics

• Similar to bar graphs, but for quantitative data.
• Bars represent specific bins or ranges of data.
• Bars have a natural ordering, usually least to greatest, with no gaps.
• Bars touch one another, unlike bar charts.
• The vertical axis can represent frequencies or relative frequencies.
• Frequencies are counts, while relative frequencies are proportions or percentages.

Example

A doctor collected cholesterol levels from a random sample of 150 patients.

• The horizontal axis represents cholesterol levels (ranges).
• The vertical axis represents relative frequencies.

Analyzing the Histogram

• Relative Frequencies:

  • First bar: 0.10 (10%)
  • Second bar: 0.15 (15%)
  • Third bar: 0.20 (20%)
  • Fourth bar: 0.30 (30%)
  • Fifth bar: 0.20 (20%)
  • Sixth bar: 0.05 (5%)

• The sum of relative frequencies should equal 1 or 100%.

  $(10\% + 15\% + 20\% + 30\% + 20\% + 5\% = 100\%)$

Determining Percentages

• To find the percentage of patients with cholesterol levels of 210 or higher, add the relative frequencies of the last three bars.
  $(0.30 + 0.20 + 0.05 = 0.55)$
• 55% of patients had cholesterol levels of 210 or higher.

Calculating the Number of Patients

• To find the number of patients in a category, use the formula:

  $Part = Percent \times Total$

• Example: Calculate the number of patients with cholesterol levels of 210 or higher.

  $Number = 0.55 \times 150 = 82.5$

• Round to the nearest whole number: Approximately 83 patients.

Practice Problem

• Cholesterol level less than 200 considered ideal.
• How many of the 150 patients had cholesterol level in this ideal range?

Solution

• Identify the relevant range: cholesterol levels less than 200 (the first group).
• The relative frequency for this group is 0.10 (10%).
• Calculate the number of patients: $(0.10 \times 150 = 15)$
• Therefore, 15 patients have cholesterol levels in the ideal range.

Key Takeaways

• Distinguish between frequencies and relative frequencies on graphs.
• In pie charts, you typically only have relative frequencies.
• Bar graphs and histograms can show either frequencies or relative frequencies.
• If you have relative frequencies, you need the total to find the number of items in a category.