Frequency and Histograms
- The horizontal axis shows the range of numbers.
- The vertical axis (frequency) represents the amount of data in each range.
The range of numbers depends on the data used.
Tutor-style math help
Frequency and Histograms: what to notice and how to work it
Statistics skill
Statistics is about describing data honestly. A calculation is useful only when it helps explain center, spread, shape, or association.
What to notice first
Ask what the display is trying to show. Histograms show shape, box plots show spread by quartiles, and scatter plots show association.
Common student mistake
Do not report a number without context. A mean, median, IQR, or regression line should answer a question about the data.
Key formulas and cues
\(\text{mean}=\frac{\text{sum}}{\text{number of values}}\)
\(IQR=Q_3-Q_1\)
\(z=\frac{x-\mu}{\sigma}\)
\(\text{slope of best-fit line}=\frac{\text{change in prediction}}{\text{change in }x}\)
A reliable path
- Identify the questionDecide whether you need center, spread, shape, or association.
- Use the right displayChoose a histogram, box plot, scatter plot, or summary statistic.
- Write the meaningExplain what the statistic says about the data set.
Worked examples
Find IQR
Example: \(Q_1=8\), median \(=12\), \(Q_3=20\)
- IQR measures the middle 50%.
- Subtract Q1 from Q3.
- 20 – 8 = 12.
Answer: \(IQR=12\)
Read association
Example: A scatter plot rises from left to right.
- As x increases, y tends to increase.
- That is a positive association.
- A best-fit line should have positive slope.
Answer: Positive association
Try one before moving on
Try: Find the mean of 6, 8, and 10.
Answer: \(8\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Frequency and Histograms: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. The median is the:
2. IQR equals:
3. A positive scatter-plot association means:
Parts of a histogram:
- Title: The title describes the information in the histogram.
- \(x\)-Axis: The \(x\)-axis is the interval that represents the scale of values that the measurements fall under.
- \(y\)-Axis: The \(y\)-axis shows the number of times the values occurred in the intervals specified by the \(x\)-axis.
- Bars: The height of the bar indicates the number of times the values occurred in the interval, while the width of the bar indicates the interval covered. The width should be the same across all bars for a histogram with equal bins.
How to plot a histogram?
To make a histogram, you must follow the steps below:
- Start by marking class intervals on the \(x\)-axis and frequencies on the \(y\)-axis.
- The scale must be the same for both axes.
- Class intervals must be exclusive.
- Draw rectangles with bases as class intervals and corresponding frequencies as heights.
- A rectangle is drawn over each class interval as class limits are shown on the horizontal axis and frequencies are shown on the vertical axis.
- The height of each rectangle is proportional to the frequency of the corresponding class if the intervals are equal.
- If the intervals are unequal, the area of each rectangle is proportional to the corresponding class frequency.
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Frequency histogram
A frequency histogram is a histogram that shows the frequency (number of occurrences) of given items.
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