From a Data List to a Histogram
In Relative Frequency and Density Histograms, you learned what the vertical scale represents in those histograms. This tutorial focuses on using a graphing calculator to make a frequency histogram from raw quantitative data, then sketching the result clearly. A calculator can count observations in bins quickly, but it cannot decide for you whether the bins and viewing window are useful.
The steps below use a TI-84-style calculator. Other graphing calculators have similar features, although menu names may differ. Enter each observation in a list, turn on a histogram plot, and specify the horizontal window. In a TI-84-style histogram, the starting value \(X_{\min}\) and class width \(X_{\text{scl}}\) determine the bin boundaries. Set the horizontal viewing range to show all the bins you want to display. The vertical window controls how much of the bars and frequency scale you can see.
A frequency histogram uses counts on the vertical axis. The bar over an interval has height equal to the number of data values in that interval. As in Constructing a Frequency Histogram, bins must cover the values you want to display, and each observation must be counted once. Touching bars are appropriate because the bins represent consecutive numerical intervals, not separate categories.
Set Up the Calculator
On a TI-84-style calculator, press STAT and choose EDIT to enter the observations in a list, such as \(L1\). Then open the statistical plot settings, turn on Plot1, and select the histogram icon. Set the histogram’s data list to \(L1\) and its frequency to 1 when each list entry is one observation. Finally, adjust the window and graph.
For a histogram, a window usually needs to show the data range horizontally and enough vertical space for the tallest bar. The following settings are especially useful:
| Setting | What it controls | Typical choice |
|---|---|---|
| \(X_{\min}\) | The left boundary where the histogram bins begin | A convenient value at or below the minimum observation |
| \(X_{\max}\) | The right side of the horizontal viewing window | A value that shows all intended bins |
| \(X_{\text{scl}}\) | The width of each histogram bin | A reasonable, easy-to-read class width |
| \(Y_{\min}\) | The bottom of the vertical viewing window | Usually 0 for a frequency histogram |
| \(Y_{\max}\) | The top of the vertical viewing window | A little higher than the tallest frequency |
| \(Y_{\text{scl}}\) | The spacing between vertical-axis tick marks | A convenient count increment |
The symbol \(X_{\text{scl}}\) can be confusing: in the histogram settings it sets the bin width, not merely the number labels you happen to print on the horizontal axis. If \(X_{\min}=0\) and \(X_{\text{scl}}=10\), the bins begin at 0 and have width 10: \([0,10)\), \([10,20)\), and so on. Set \(X_{\max}\) far enough to display the last intended bin. If the calculator offers automatic window settings, they can be a starting point, but check that the resulting boundaries and scales suit your question.
When you sketch the result, include a title or contextual description, label the horizontal axis with the variable and units, and label the vertical axis “Frequency” or “Count.” Show the class boundaries, a useful frequency scale, and touching bars at the correct heights. A sketch is not just a row of approximate rectangles: its labels and scales should make the meaning of the display clear.
Worked Example: Enter Data and Sketch a Histogram
Worked Example: Enter Data and Sketch a Histogram
A fictional group of 20 students records how many minutes they spend reading on a particular evening. Enter these values into a calculator and sketch a frequency histogram with 10-minute bins:
\(4, 7, 9, 11, 13, 15, 16, 18, 19, 19, 21, 22, 24, 26, 27, 29, 31, 34, 38, 43\)
Enter and select the data. Put the 20 observations in \(L1\). Turn on Plot1, choose the histogram plot type, select \(L1\) as the data list, and set the frequency to 1. That tells the calculator to count each entered value once.
Choose boundaries and a window. The smallest observation is 4 and the largest is 43. Set \(X_{\min}=0\), \(X_{\max}=50\), and \(X_{\text{scl}}=10\), giving bins \([0,10)\), \([10,20)\), \([20,30)\), \([30,40)\), and \([40,50)\). Set \(Y_{\min}=0\), \(Y_{\max}=8\), and \(Y_{\text{scl}}=2\), so the vertical window starts at zero and has room for a bar as high as 8.
Check the bar heights. Count the data values in each interval. For example, \(4,7,9\) fall in \([0,10)\), and \(11,13,15,16,18,19,19\) fall in \([10,20)\). The complete frequency table is:
| Reading time (minutes) | Frequency |
|---|---|
| [0, 10) | 3 |
| [10, 20) | 7 |
| [20, 30) | 6 |
| [30, 40) | 3 |
| [40, 50) | 1 |
| Total | 20 |
The frequencies add to \(3+7+6+3+1=20\), matching the number of observations entered. The calculator should show five touching bars with those heights. To sketch the result, label the horizontal axis “Reading time (minutes)” and mark 0, 10, 20, 30, 40, and 50. Label the vertical axis “Frequency” and mark counts from 0 to 8, perhaps in steps of 2. Draw bars over the five intervals at heights 3, 7, 6, 3, and 1, respectively. In context, the tallest bar shows that the most students in this group reported reading for at least 10 but less than 20 minutes.
Use the Window to Control What You See
A viewing window and a bin definition are related but not identical. Changing \(X_{\text{scl}}\) changes the bin width and can change the shape of the histogram. Changing \(Y_{\max}\), by contrast, changes how much vertical space is visible; it does not change the counts in the bins. If \(Y_{\max}\) is below the tallest bar, the graph is clipped and the bar height cannot be read accurately. If it is unnecessarily large, the bars may look very short. A vertical scale beginning above zero can also make frequencies harder to judge, so start at zero for a frequency histogram.
The horizontal range should not cut off observations or bins that matter to the display. If the minimum or maximum values lie outside the visible range, some bars may be missing or partly hidden. When you revise the horizontal window, check that the starting boundary and bin width still produce the intervals you intend. Do not change a bin setting accidentally just to make the graph fit the screen.
Worked Example: Change the Bin Width and Compare
Worked Example: Change the Bin Width and Compare
A fictional group of 16 commuters records travel time to school, in minutes: \(3,5,8,9,11,12,14,16,18,21,22,24,27,31,35,37\). Make a histogram using 10-minute bins, then change to 20-minute bins. Explain what changes in the graph.
Set the 10-minute histogram. Enter the values in \(L1\), choose a histogram of \(L1\) with frequency 1, and use \(X_{\min}=0\), \(X_{\max}=40\), and \(X_{\text{scl}}=10\). The bins and counts are \([0,10):4\), \([10,20):5\), \([20,30):4\), and \([30,40):3\). These counts total \(4+5+4+3=16\). Set \(Y_{\min}=0\), \(Y_{\max}=6\), and \(Y_{\text{scl}}=1\), which leaves room to see the tallest bar of height 5.
Change only the bin width. Keep the same data list and set \(X_{\text{scl}}=20\), with \(X_{\min}=0\) and \(X_{\max}=40\). The new bins are \([0,20)\) and \([20,40)\). The first contains the four values below 10 and the five values from 10 up to 20, for a total of 9. The second contains the four values from 20 up to 30 and the three values from 30 up to 40, for a total of 7. The two new bar heights are 9 and 7; they still total 16.
Sketch and interpret the change. With 10-minute bins, draw four touching bars with heights 4, 5, 4, and 3 over consecutive intervals from 0 to 40. With 20-minute bins, draw two touching bars with heights 9 and 7 over \([0,20)\) and \([20,40)\). The wider bins combine neighboring counts, so the second histogram has fewer bars and shows less detail. Both displays describe the same 16 travel times; the choice of bin width changes the visible pattern, not the observations.
Check the Calculator’s Graph Before You Copy It
The calculator’s display is a tool for counting and visualizing, not a substitute for checking the setup. Confirm that the selected list contains the intended variable, that the frequency setting counts observations appropriately, and that the bin boundaries cover the data. Then compare the bar heights with a quick frequency count or table. A mismatch can reveal an incorrect list, an unsuitable window, or a setting you did not mean to change.
For raw observations, keep the frequency setting at 1 so each list entry contributes one observation. A calculator may allow a separate frequency list that weights each data value. That feature is not the same as entering a frequency table of bin counts: feeding bin counts as weights generally repeats individual values, rather than telling the calculator how many observations fall throughout each interval. If all you have is a grouped frequency table, the exact individual observations are not available, so you cannot recreate the original raw-data histogram precisely from that table alone.
Worked Example: Set a Window and Verify the Output
Worked Example: Set a Window and Verify the Output
A fictional greenhouse records the heights, in centimeters, of 12 seedlings: \(12,14,16,17,18,21,24,26,28,29,31,33\). Create and check a histogram with 5-centimeter bins from 10 to 35 centimeters.
Enter the observations and set the plot. Enter all 12 heights in \(L1\); select a histogram using \(L1\) and frequency 1. Set \(X_{\min}=10\), \(X_{\max}=35\), and \(X_{\text{scl}}=5\). These settings create the intervals \([10,15)\), \([15,20)\), \([20,25)\), \([25,30)\), and \([30,35)\). Use \(Y_{\min}=0\), \(Y_{\max}=4\), and \(Y_{\text{scl}}=1\).
Verify each count from the list. The values 12 and 14 are in \([10,15)\), giving 2. The values 16, 17, and 18 are in \([15,20)\), giving 3. The values 21 and 24 are in \([20,25)\), giving 2. The values 26, 28, and 29 are in \([25,30)\), giving 3. Finally, 31 and 33 are in \([30,35)\), giving 2. Thus the bar heights should be 2, 3, 2, 3, and 2, and they add to \(2+3+2+3+2=12\).
Sketch and label the display. Draw five touching bars across the horizontal range 10 to 35, with those heights in order. Label the axes “Seedling height (centimeters)” and “Frequency.” The bars alternate between heights 2 and 3. In this sample, 3 of the 12 seedlings are at least 15 but less than 20 centimeters tall, and 3 are at least 25 but less than 30 centimeters tall.
Common Mistakes and AP Exam Tips
- Plotting the wrong list. Check the selected data list before interpreting the graph. A correctly drawn histogram of the wrong variable does not answer the question.
- Using an unhelpful starting boundary or bin width. \(X_{\min}\) and \(X_{\text{scl}}\) determine where bins start and how wide they are. State or show the interval boundaries you intend, then confirm the calculator uses them.
- Clipping a bar. If \(Y_{\max}\) is too small, a bar can extend beyond the screen. Increase \(Y_{\max}\) so the tallest bar is fully visible; do not mistake the visible portion for its frequency.
- Leaving out values at the ends. Check that the horizontal range includes the observations and intervals you need. A useful starting point is to set the window to cover the minimum and maximum values, with convenient boundaries.
- Making bars separated. A histogram displays consecutive numerical intervals, so its bars touch. Separated bars are used for categorical data in a bar chart.
- Copying the screen without labels or a scale. A full-credit sketch makes the variable, units, intervals, and frequencies clear. Include a title or contextual description and mark a readable vertical frequency scale.
For a strong response, name the variable and units, identify the chosen bins, and show or report the frequency in each bin. If asked to sketch, reproduce the bar heights and boundaries accurately and label both axes. A calculator can draw the bars, but your explanation should still tell the reader what those bars represent.
Check Your Understanding
For each question, think about both the calculator settings and the meaning of the resulting histogram.
- A list contains 18 observations, and each entry is one person’s measurement. What frequency setting should you use so each observation counts once?
- If \(X_{\min}=5\) and \(X_{\text{scl}}=4\), what are the first three bin intervals?
- A histogram has a tallest bar of frequency 11, but \(Y_{\max}=8\). What is wrong with the display, and what should you change?
- Why can two histograms of the same data have different shapes when their bin widths differ?
- A frequency table gives the counts in five intervals but not the original observations. Why might entering those five counts as a calculator frequency list fail to reproduce the original histogram?