0580 · 9.4Statistical charts and diagrams

Frequency tables and bar charts

0:00A list of forty journeys to school can be difficult to understand at a glance. A clear chart can reveal the most common journey immediately. But different diagrams reveal different features of the same information. In this video, we will turn data into useful pictures, then learn how to read those pictures accurately. The central idea is that every diagram represents quantities using something visible. A bar chart uses length, while a pie chart uses angles within a circle.

0:25A pictogram uses symbols, and a stem-and-leaf diagram organises individual values. Understanding what each feature represents makes these diagrams much easier to draw and interpret. For our first example, imagine these results from a class survey. Each student chooses one usual method of travelling to school. The categories are walking, bus, cycling and car. Frequency means the number of times a category or value occurs, so sixteen students belong to the walking category.

0:52The total frequency tells us how many observations are represented. Adding sixteen and twelve gives twenty-eight. Including the eight cyclists brings the total to thirty-six, and adding four car passengers gives forty. Checking this total helps us avoid losing or counting an observation twice. A simple frequency distribution lists the possible values or categories alongside their frequencies. It summarises a collection without changing how many observations there are. You need basic counting, addition and fractions to work with these displays.

Comparing bar chart types

1:22For pie charts, you also need to know that a complete turn contains three hundred and sixty degrees. A bar chart makes frequencies visible through bar heights. Here, walking has the tallest bar because it has the greatest frequency. The categories occupy separate positions along the horizontal axis. Gaps between bars make that separation clear, and equal widths prevent a category from looking more important simply because its bar is wider.

1:46To draw this chart, first label the categories and the frequency axis. Choose an evenly spaced scale that reaches at least the largest frequency. Our scale increases by two at each marked step and starts at zero. Draw equal-width bars reaching sixteen, twelve, eight and four, with consistent gaps between them. Reading a bar means matching its top to the vertical scale. The bus bar reaches twelve, even though it is only the second bar.

2:11Its position identifies the category, not its frequency. Walking exceeds cycling by 16 − 8, giving eight students, so the height difference represents a difference in counts. Always inspect the scale before judging a chart visually. If each grid division represents two students, a height of six divisions means twelve students. A frequency axis that starts above zero can exaggerate differences. When drawing these straightforward bar charts, use a zero baseline so bar lengths represent frequencies fairly.

2:38A dual bar chart places two bars side by side for each category. The key tells us which colour represents each group. This makes comparisons within a category straightforward because both bars begin at the same baseline. Here, we can compare the travelling habits of two classes without confusing their frequencies. For an exam-style interpretation, determine how many more students use the bus in Class Y than in Class X.

3:01Read fourteen from the Class Y bar. Subtract the Class X frequency of twelve to obtain two. The answer is two students, not the difference between the positions of the bars. A composite bar chart, also called a stacked bar chart, places the groups on top of one another. The total height gives the combined frequency. Each individual section has a height equal to its own group frequency. This shows totals clearly, but an upper section must be measured between its bottom and top.

3:27On the stacked walking bar, the top reaches twenty-six and the colour changes at sixteen. The upper section therefore represents 26 − 16, which equals ten students. Reading twenty-six as the Class Y frequency would include both classes. Side-by-side bars emphasise comparisons, while stacked bars emphasise how separate parts contribute to a total.

Calculating pie chart angles

3:47A pie chart represents the whole collection as one complete circle. Each category receives the same fraction of the circle as its fraction of the total frequency. Walking accounts for sixteen out of forty students. Its sector must therefore contain 16/40 of the circle’s three hundred and sixty degrees. To draw the pie chart, first divide three hundred and sixty degrees by the total frequency of forty. This gives nine degrees per student.

4:12Multiplying sixteen by nine gives a walking angle of one hundred and forty-four degrees. The same allocation of nine degrees per student applies to every category. For the bus sector, 12 × 9 gives one hundred and eight degrees. The cycling angle comes from 8 × 9, giving seventy-two degrees. Four car passengers require thirty-six degrees. Adding all four angles gives three hundred and sixty degrees, confirming that the sectors fill exactly one circle.

4:37Draw a circle with a compass and mark its centre. Draw a starting radius, then use a protractor to measure the first sector at the centre. Continue from the latest boundary for each remaining angle. Label the sectors clearly, using either category names or a key that identifies every colour or pattern. The general calculation is frequency divided by total frequency, multiplied by three hundred and sixty degrees. Dividing first finds the fraction of the whole.

5:03Multiplying by the full-circle angle converts that fraction into an angle. Do not divide by the number of categories unless every category genuinely has the same frequency. You can also work backwards from an angle to a frequency. A sector of seventy-two degrees occupies 72/360 of the circle. That fraction simplifies to one fifth. With forty students altogether, ⅕ × 40 gives eight students in that category. Pie charts show proportions, not totals on their own.

5:31Equal-sized sectors in two charts need not represent equal numbers if the surveys have different totals. Always use the stated total when calculating a frequency. A complete pie chart must represent categories that together account for the whole collection. Here is an exam-style interpretation question. This pie chart shows the favourite fruit of sixty students, but the grapes angle is missing. The banana sector measures one hundred and twenty degrees.

5:53Orange brings the running total to two hundred and sixteen degrees, and apple takes it to three hundred degrees. Subtracting from three hundred and sixty degrees leaves sixty degrees for grapes. Next, find how many students chose apple. Dividing three hundred and sixty degrees by sixty students gives six degrees per student. The apple angle of eighty-four degrees, divided by six, gives fourteen students. Grapes, with sixty degrees, represents ten students, and all four frequencies together make sixty.

Pictograms and stem-and-leaf diagrams

6:21A pictogram represents frequency using repeated symbols. The key tells us how many observations each symbol represents. In this chart, one complete circle stands for four students. Four walking symbols therefore represent sixteen students, while the single car symbol represents four rather than one. To draw a pictogram, choose a useful key and divide each frequency by the number represented by one symbol. Keep all complete symbols the same size.

6:46Enlarging one symbol changes its appearance without clearly communicating a frequency. Arrange the symbols neatly so rows are easy to compare, and always include the key. Suppose another survey records ten cyclists, using the same key. Divide ten by four to obtain two point five symbols. Two complete symbols represent eight students. Subtracting eight from ten leaves two students, so we draw half a symbol for the remainder. A partial symbol represents the corresponding fraction of the key value.

7:13Half a symbol here means two students, not half a student. When interpreting a pictogram, count complete and partial symbols before applying the key. Missing the key is a common reason for an otherwise careful answer to be wrong. A stem-and-leaf diagram organises numerical data while keeping every original value visible. Consider the eleven quiz scores on the board. For these two-digit whole numbers, the tens digit becomes the stem and the units digit becomes the leaf.

7:38A score of twenty-four therefore contributes a leaf of four beside stem two. To construct the diagram, arrange the scores in ascending order. The teens produce three leaves beside stem one. The twenties contribute five leaves beside stem two, and the thirties supply three beside stem three. Adding those leaf counts gives eleven, matching the number of scores we started with. Each row must have its leaves in ascending order.

8:02Repeated values need repeated leaves because each leaf represents one observation. The two scores of twenty-four therefore appear as two separate fours. Include the key stating that two bar four means twenty-four marks, so the place value is unambiguous. The zero beside stem three is essential because it represents the score of thirty. Without it, one observation would disappear. Reading the first and last entries shows that the smallest score is seventeen and the largest is thirty-two.

8:27Counting the leaves beside stem two tells us that five scores are in the twenties. Unlike a bar chart, the horizontal position of a leaf is not a measurement scale. Leaves simply form an ordered list within each stem. The key determines how stems and leaves combine, especially when decimal values are used. Never assume the units without checking what the key says.

Grouped frequencies and checks

8:47A simple frequency distribution can also list single whole-number values. This table shows how many pets each of twenty students owns. Five students have no pets, and eight have exactly one. To find how many own at least two, add the frequencies for two and three, which gives seven students. A frequency distribution can also collect numerical values into groups. Here, the first class runs from ten up to and including twenty.

9:11The board writes this class as x greater than ten and less than or equal to twenty. The second class contains scores greater than twenty and up to and including thirty, so thirty belongs in that second class. The first class contains three of our scores. Including thirty in the second class gives six observations there. The last class contains the two scores of thirty-two. Their frequencies add to eleven, but this grouped table no longer reveals every individual score as the stem-and-leaf diagram does.

9:38At this level, questions can ask you to draw or interpret any of these displays. Read the title, labels, scale and key before calculating. For stacked bars, identify the boundaries of the required section. For pie charts, establish the total frequency before converting between angles and numbers. These methods must work without a calculator on Paper Two. Paper Four allows a calculator, but the meaning of the scale or key remains just as important.

10:04Keep intermediate calculations unrounded, especially when finding pie chart angles. Follow any stated accuracy instruction, with answers that are not exact normally given to three significant figures unless otherwise specified. The strongest final check is to connect the diagram back to the data. Bar heights should match the frequencies, and pie chart angles should total three hundred and sixty degrees. Pictogram symbols must agree with their key. A stem-and-leaf diagram should contain one leaf for every observation, including repeats and zeros.

10:32Each display answers a slightly different need. Bar charts compare quantities, dual bars compare groups, and stacked bars show how parts combine. Pie charts show shares of a whole, while pictograms encode counts through a key. Stem-and-leaf diagrams preserve individual values, and frequency distributions organise how often categories or values occur.