Reading Data and Graphs

From raw data to a frequency table

The observations you collect before any sorting are the raw data, whether or not they are numbers. Imagine asking $22$ students which pet they like best. You write down each answer as it arrives: dog, cat, dog, bird, fish, dog, and so on. That list is complete but hard to use. The first step in reading data is to organize it into a frequency table. That table lists each category beside its frequency, the number of times that category occurs. A quick way to build one is to make a tally mark for each answer and then count the marks.

Two different kinds of thing live in this table, and keeping them straight is the whole skill. The category is what was chosen, the pet, and the frequency is how many students chose it. The frequency of dog is $8$; “dog” itself is not a number at all. Notice also that the frequencies add up to the total number of students,

because every student was counted in exactly one category. That “the parts add to the whole” fact comes back when we reach circle graphs. The pictures below are all built from data organized this way, though not every one counts frequencies the way this table does: a bar graph or pictograph usually does, while a line graph instead follows a measurement as it changes over time.

Pictographs: a key gives each symbol a value

A pictograph (or picture graph) shows each category’s value as a row of repeated symbols, a frequency for the pet survey below. By itself a symbol means nothing; what gives it meaning is the key, a short note that says how many each symbol stands for. To read a value, count the symbols in a row and multiply by the key. A partial symbol stands for that same fraction of the key, so half a symbol is half the key’s value.

Reading the Bird row shows why the key matters: there are two full circles and one half circle, which is $2.5$ symbols. So the frequency is $2.5 \times 2 = 5$, exactly the $5$ from the table. Someone who ignores the key and just counts “two and a half pictures” reports the wrong number. Someone who rounds the half circle up to a whole reports $6$ instead. The picture is friendly, but the key is doing the real work.

Bar graphs: comparing categories at a glance

A bar graph draws each category’s value as a bar, where the length of the bar represents that value, a frequency for the pet survey, but sometimes a different kind of measurement. The bars all have the same width and sit apart from one another, so only their lengths carry meaning. To read one bar’s value, follow its end across to the number scale. To compare categories, just compare the bar lengths directly, since the longest bar shows the greatest value.

One feature of this scale is easy to take for granted but important: it starts at $0$. Because the bottom of every bar is $0$, a bar that reaches $8$ really is twice as tall as a bar that reaches $4$, so the heights are in the same ratio as the values. That shared baseline is what makes a bar’s height a fair stand-in for its value: a longer bar reads as a proportionally bigger number, not just a bigger one. You can still read an exact value, or find the difference between two values, straight off the labeled scale even when the axis does not start at $0$. What a cut-off axis breaks is judging size by eye, from the bar’s height alone. Later you will see how a scale that does not start at $0$ can make a small difference look far bigger than it is.

When you want to compare two groups across the same categories, you can draw a double bar graph, also called a grouped bar graph. It puts two bars side by side for each category, one per group, with a legend telling you which is which. Reading it is the same skill twice. Pick the category, then read the bar for the group you care about.

Line graphs: showing change over time

A line graph is built for a quantity that changes over time. Each reading is plotted as a point, with time running along the horizontal axis and the quantity up the vertical axis. The points are then joined left to right by straight segments. The segments are what make a line graph special: a segment that rises means the quantity went up between those two times. A segment that falls means the quantity went down between those two times, and a flat segment means it did not change. When the readings are evenly spaced in time, as they are below, the steeper the segment, the faster the change.

This is also why a line graph is the wrong tool for separate categories like the pets above. A segment between two points shows the direction and size of the change between those two readings. It does not claim that a value was measured in between, or even that a meaningful in-between value exists: the museum graph above has no meaningful visitor count halfway between Monday and Tuesday, yet the segment there still usefully shows that the count rose. What a segment does need is an axis with a genuine order, almost always time, so that “direction of change” means something. Dog and Cat have no such order: swapping their positions on the axis would be just as valid, so a segment joining them would trace a trend that depends on an arbitrary arrangement, not a real one.

Circle graphs: parts of a whole

A circle graph (or pie graph) shows how a whole splits into parts. The entire circle stands for the whole data set, all $100\%$ of it, and each category is a slice whose size is that category’s share. A bigger share is a bigger slice, so a half-circle slice is half the data and a quarter-circle slice is a quarter.

Every student is counted in exactly one travel group, so the shares can’t miss anyone and can’t double count anyone: added together, they rebuild the whole group. In the travel graph below, $50\% + 25\% + 15\% + 10\% = 100\%$, the same “parts rebuild the whole” idea you already checked with the frequencies $8 + 6 + 5 + 3 = 22$ at the start of this lesson.

To read a circle graph, read each slice’s labeled share, usually a percent. The percents tell you the proportions directly: the biggest slice is the most common category, and a $50\%$ slice is half of everything. To turn a share into an actual count, multiply the percent by the total. If $240$ students are shown and the “walk” slice is $25\%$, then the number who walk is $\frac{25}{100} \times 240 = 60$.

Choosing the right display

Reading is easier when the data is shown the right way in the first place, so it helps to know what each display is for:

• A bar graph or pictograph compares amounts across separate categories, like favorite pets or books per student. The bars or symbol counts make the comparison obvious.
• A line graph shows how one quantity changes over time, like temperature through a day. Use it only when the horizontal axis is ordered, so the connecting segments mean something.
• A circle graph shows how a whole splits into parts, like the share of a budget spent on each category. It is best when you care about proportions rather than exact counts.
• A table of exact numbers, a frequency table for categories or a table of measurements over time, sits behind every one of these pictures. Reach for it when you need a precise value rather than a quick picture.

A display used outside its purpose misleads even when every number is correct. A line graph of unrelated categories invents a trend that is not there. A circle graph only makes sense when the slices really are parts of one whole.

Reading a graph honestly

Reading a graph well also means not reading in more than it says. Two habits keep you honest.

First, draw only conclusions the data supports. A graph reports what was measured, not why. If ice cream sales and sunburns both rise over the summer, a graph of the two does not show that ice cream causes sunburns; a graph never reveals what actually caused a rise, only that one happened. And a line graph stops where the data stops, so reading a value past the last plotted point is a guess, not a reading.

Second, check the scale before you trust a comparison. The fair-comparison promise of a bar graph depends on the value axis starting at $0$. When it does not, the bar lengths are no longer proportional to the values, and small differences look huge.

In the figure, Brand A’s bar towers over Brand B’s, yet the true values are $52$ and $48$, a difference of just $4$. Reading the numbers off the scale, rather than eyeballing the bar heights, is what protects you. A truncated scale is not always dishonest, but it is always worth noticing.

Common mistakes