C10.5; E10.5 - Statistics on a calculator
A calculator can give a precise-looking mean for the wrong list. How can you tell whether it has understood your frequency table? C10.5 and E10.5 connect data entry to mean, median and quartiles, then use midpoints to estimate a grouped mean on a graphic display calculator.
Data entry
The observations describe five measurements. The shorter table “2 occurs once, 3 twice, 5 once, 7 once” describes exactly the same data. Frequencies tell the calculator how often to count a value; they are not extra measurements.
Value and frequency lists
A value list contains the measured values. Its paired frequency list contains their counts. One-variable statistics must use both lists when observations are entered in frequency form. The displayed count should equal the sum of the frequencies.
Menu names differ between calculators. Use the statistics/data editor, clear old lists, select the intended data list and explicitly check whether frequency is set to 1 or to a frequency list. A blank row, old entry or remembered frequency setting can change the calculation.
Worked example 1: entering a raw list
Question. Enter for one-variable statistics and state two checks on the entry.
Plan. Enter all five observations in one value list and use frequency 1 for each row.
| Value list | 2 | 3 | 3 | 5 | 7 |
|---|---|---|---|---|---|
| Frequency setting | 1 | 1 | 1 | 1 | 1 |
Select that value list for one-variable statistics. Read the displayed count and, if available, the sum of values.
Answer. and the sum is 20.
Check. The written list contains five numbers, and . Matching both catches more mistakes than matching the count alone.
Now repeated observations are compressed into counts. The number of rows is no longer the number of observations.
Worked example 2: entering a frequency table
Question. Numbers of calls 0, 1, 2 and 3 have frequencies 1, 2, 3 and 4. Describe the correct entry and check its total count.
Plan. Put call counts in the value list and their frequencies in a second list, then link that list as frequency.
| Value list | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Frequency list | 1 | 2 | 3 | 4 |
Select the first list as data and the second as frequency. Each row now represents its stated number of observations.
Answer. The calculator should report .
Check. Expanding the table gives , ten values with sum 20. A count of 4 would reveal that frequency had been left at 1.
Return to the opening problem: the count is the first quick test that the calculator interpreted the table properly. Also compare the smallest/largest data values with the table where these are displayed.
Averages and quartiles
Once entry is correct, use the permitted one-variable summary statistics to read the mean, median, lower quartile and upper quartile. Common display labels are , Med, and . Read the label, rather than selecting a number because it appears first.
Summary statistics
The mean uses all the values; the median is the middle of ordered data. The lower and upper quartiles describe the quarter and three-quarter positions. A calculator's quartile convention can differ from a hand convention on some short lists: use the required calculator output when asked to use a GDC.
These examples use repeated values around the quartile positions so that the usual calculator conventions agree. Quartiles must satisfy .
Worked example 3: reading raw-data summaries
Question. Use a GDC to find the mean, median and quartiles of .
Plan. Enter eight raw values with frequency 1, run one-variable statistics and read the four labelled outputs.
The summary gives
Answer. Mean 3, median 3, lower quartile 2 and upper quartile 4.
Check. Pairs from opposite ends each sum to 6, so the mean is 3. The middle values are 2 and 4, averaging to 3; the central values in the lower and upper halves are 2 and 4.
Continue this lesson.
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