Frequency Distribution Graphs

1Summary 重點摘要

Learn how to make frequency distribution table and stem-and-leaf diagram, and how to read them.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Why group dataA handful of values can be listed one by one, but with much data (the heights of 40 students, a month of daily temperatures) a raw list is long and patternless, so the data are grouped. For class heights ranging from 140 cm to 180 cm, take 10 cm per class: 140–149, 150–159, 160–169, 170–179, then tally the count in each. Grouping compresses the information into a few classes plus frequencies, making the shape obvious at a glance: which height range holds the most students and which the fewest. Grouping is the first step in organising mass data and the foundation of every statistical chart.
2. Step 2: The frequency distribution tableA frequency distribution table lists each class together with its frequency (the number of data inside). For the 40 students' heights: 140–149 cm holds 5 students, 150–159 cm holds 15, 160–169 cm holds 12, and 170–179 cm holds 8; the total 5 + 15 + 12 + 8 = 40 matches the class size — an essential check. Two points when constructing the table: classes must neither overlap nor leave gaps (no value may fall between two classes), and the table needs a title with units (such as 'height (cm)') so readers know what each column means.
3. Step 3: The stem-and-leaf diagramThe stem-and-leaf diagram is a statistical chart that keeps the raw data: each value splits into a 'stem' (the tens digit) and a 'leaf' (the units digit). For the data 12, 15, 18, 23, 25, 31: stem 1 carries leaves 2, 5, 8; stem 2 carries leaves 3, 5; stem 3 carries leaf 1, arranged as
   
   1 | 2 5 8
   2 | 3 5
   3 | 1
   
   with every stem's leaves sorted from small to large. The picture shows at once that most values lie in 10 to 19. Compared with a grouped table, the diagram's advantage is that every original value can still be read back — something a frequency table cannot offer.
4. Step 4: Benefits of stem-and-leafThe stem-and-leaf diagram offers three benefits. First, the shape is visible: where the data cluster, whether they are symmetric, whether outliers stick out — all at a glance. Second, the raw data survive: the median, mode, maximum and minimum can be read straight off the diagram; in the one above the minimum is 12, the maximum 31 and the median 20.5. Third, two data sets compare easily: place two diagrams back to back sharing a central stem, and two classes' exam scores or two teams' heights can be compared instantly. These features make the diagram a tool that is table and chart at once.
5. Step 5: How to read a stem-and-leafReading a stem-and-leaf diagram starts with the key, because the key fixes the place value. If the key states 1 | 2 = 12, stems are tens and leaves units, so 2 | 3 is 23. But if the key states 1 | 2 = 1.2, the very same diagram means something entirely different and 2 | 3 becomes 2.3. Without a key the diagram cannot be read uniquely, and examiners deduct marks for its absence. Suggested reading order: read the key first, then the values stem by stem, and finally find whatever the question asks — maximum, minimum, median or mode. The habit of 'key first' prevents misreading the whole question.
6. Step 6: Answering questions from the tableA frequency table answers many questions directly. Using the table above (140–149: 5; 150–159: 15; 160–169: 12; 170–179: 8): how many students measure 150 cm or more? The three classes 150–159, 160–169 and 170–179 total 15 + 12 + 8 = 35. The modal class — the one with the most students — is 150–159 cm with 15. As a share of the class, the tallest group is 8 ÷ 40 × 100% = 20%. Note that grouping reveals only 'how many per class', never how many sit at one exact height — the price of grouping, and precisely the value of a stem-and-leaf diagram that keeps every value.
7. Step 7: Exam tips and common mistakesFour slips cost marks in exam stem-and-leaf and table questions. First, omitting the key — always add one, such as 2 | 5 = 25. Second, unsorted leaves: 1 | 5 2 8 must be written 1 | 2 5 8; disorder loses marks. Third, overlapping class limits: writing 140–150 and 150–160 leaves a student of exactly 150 cm with no home; use 140–149 and 150–159 for integer data, or state the boundaries explicitly. Fourth, forgetting to confirm that the frequencies sum to the total number of data. Ten seconds checking these four items secures the marks of the whole statistics question.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Group when data is many
• Frequency table: how many in each group
• Stem-and-leaf: tens as stem, units as leaf
• Must add key
• Leaves must be sorted