In the previous lesson, you learned that quantitative data is information describing the extent or quantity of a certain measurement and uses a number and a unit. The number tells us how many (or how much), and the unit tells us what the scale of measurement is. Quantitative data can be divided into two categories. Study the tabs to learn about each one.
Exact numbers are numbers that have a value that is exactly known. Exact numbers are either defined numbers within a given measurement system, or those that are obtained by counting individual objects. An exact value can be expressed in only one way and is not simplified or rounded. We can be certain of the quantity of an exact value.
For example, a dozen is defined as exactly 12; you can be certain a carton with a dozen eggs contains 12 eggs inside because the value of a dozen is defined.
In addition, there are exactly two chairs in this image. The number of chairs is counted, not measured, so we are completely certain how many chairs there are.
For Example
Examples of exact values include data such as those listed in this table.
- Conversions in the Imperial System (i.e., pounds to ounces, feet to miles, etc.)
- Conversions in the Metric System (i.e., kilograms to grams, centimeters to meters, etc.)
- The words per/percent mean exactly 100.
- Counted numbers (i.e., 15 people in a store, 34 books on a shelf, etc.)
Measured numbers are numbers obtained by a measuring instrument or device. Measuring devices are items such as scales, rulers, tape measures, thermometers, clocks, measuring cups/spoons, beakers, and graduated cylinders--to name a few. The measured value is only as good as the device that was used to measure the quantity.
For example, this scale measures the mass of the three oranges as 1.012 kg. The mass is implied to be known to the nearest thousandth of a kilogram since the scale provides three decimal places. The mass might be closer to 1.010 or 1.014 because there is some inherent uncertainty involved with any measurement. If we had used a scale that is able to detect differences in mass using more than 3 decimal places, we could have a measured value that is closer to the exact mass of the oranges, but they will still have some amount of uncertainty.
While we can be certain that there are three oranges on the scale (an exact value, because we can count them), there is some degree of uncertainty in the mass of these oranges because mass is a measured value provided by a device (scale) that has limitations.
For Example
Examples of measured values include data such those listed in this table.
- the diameter of a coin, such as 10.2 mm
- the mass of an object, such as 8.887 grams
- the length of a pen, such as 12 cm
Let's Practice
Practice distinguishing between exact numbers and measured numbers by completing this activity. Determine whether the number described on each slide is an exact number or a measured number. Then check your answer.
the number of seeds in an apple
exact number
The number of seeds is certain because they can be counted. In this example, there are five seeds.
temperature recorded by a thermometer.
measured number
Temperature is measured using an instrument, the thermometer. The value in the example is 36.1 degrees and is defined by the limit value the thermometer can measure.
There are 12 inches in every foot.
exact number
The number of inches in a foot is certain because they are defined. In this example, there are 12 inches in one foot.
the distance determined by a measuring wheel
measured number
Distance is measured using an instrument, the measuring wheel. The value in the example is defined by the limit value the wheel can measure.
time recorded by a stopwatch
measured number
Time is measured using an instrument, the stopwatch. The value in the example is 6 minutes, 30 seconds and is defined by the limit value the watch can measure.
