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Tutorial 1: Statistics PREP FOR THE AP EXAM

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cientific evidence rarely hinges on the result of a single Average: Mean, Median,
Sexperiment, measurement, or observation. As we discussed in
Module 0, any scientific claim must be backed up by data from and Mode
multiple subjects across multiple repetitions of the same exper-
imental process or multiple observations. It is the accumulation hen scientists make observations or do an experiment, they
of evidence from many independent sources, all pointing in the Wgather results and collect data. The direct observations and
same direction, that lends weight to a scientific hypothesis until measurements that have not been organized, processed, or interpreted
eventually it becomes a theory. are known as raw data. For example, a doctor collects all kinds of raw
data about your health, such as height, weight, body temperature,
This means that any statement of a number—ants can carry breathing rate, heart rate, and blood pressure. You may give a blood
10 times their weight; a human cell can divide in 24 hours—is or urine sample so that the concentrations of various substances can
actually a statement of many numbers. How do scientists deter- be measured. Or you may spit into a vial so that a DNA sample can
mine what number will represent all of their experimental results? be obtained to look for common genetic variants associated with
And how do scientists succinctly describe all of the individual disease. The data collected by physicians are all types of raw data.
data points? Statistics is a field that helps scientists to organize, After scientists collect raw data, they often process the data in
analyze, and see trends in data. In this tutorial, we will discuss sev- some way in order to make sense of it. One way to process it
eral statistical tools that help us understand and interpret data. We involves determining the average of all of the measurements.
will start with a quick look at the issue of precision in reporting Determining the average takes all of the individual pieces of data
calculations.
and provides a single, representative number. In this section, we
look at several different ways to calculate the average.
Significant Figures
Mean
hen reporting any recorded data or quantitative conclu-
Wsions, you should use appropriate precision. For example, if The first type and most common type of average is called the
you measure something and say it is 2 meters tall, does that mean mean Given a set of values, the mean is obtained by adding

.
it is exactly 2 meters or perhaps 2.03 meters or even 2.10 meters? all of the values in the dataset and dividing by the number of
Significant figures indicate the precision of a measurement. values in the dataset, which is indicated by n .
Significant figures are numbers that carry meaning. For example,

2 meters has just one significant figure because it has only one Example: In the dataset of nine values, 5, 9, 4, 6, 5, 1, 5, 9, 1, 9,

digit. By contrast, 2.03 has three significant figures. The more the mean is determined by adding all of the values (5 + 9 + 4 +

,
significant figures, the more precise the measurement. 6 + 5 + 1 + 5 + 9 + 1 + 9) = 54 and then dividing by the number
54

of values (n = 9) , or = 6 .
9
In general, all nonzero numbers (1, 2, 3, and so on) are The mean can also be calculated using the following formula:
significant. Zeros between other numbers are also significant.
n
For example, as we noted, the number 2.03 has three signifi- = 1 ∑ i
x
x
cant figures. Leading zeros (zeros that are located to the left of n i =1
another number) are not significant. The number 0.02 has only In this formula, x represents the mean. ∑ the Greek letter sigma, is

,

one significant figure. Trailing zeros (zeros located to the right n
∑

of another number) are also not significant, except in numbers a symbol that means “the sum of.” x i means x + x + … + x ,
2
1
n
i=1
with a decimal point. For example, the number 2.10 has three where x is the n th, or final, value, so it indicates that you sum all

n
1

significant figures. of the values. And indicates that you divide the result by the
n
number of values.
When doing calculations, significant figures in a final answer are
determined by the number that is the least precise in your dataset.
However, do not round intermediate values when you perform Median
calculations; only round your final answer. For example, let’s say

you are asked to calculate the area of a rectangle (width (w) × Another type of average is the median . The median is middle

height (h)) , with w = 4.33 feet and h = 2 feet. The answer is value of a group of values. That is, there are as many values falling
,

9 feet. That is: 4.33 × 2 = 8.66 but the answer is rounded up above the median as below it.

to 9 because, among the numbers you used in the calculation, a The median is sometimes useful because it is less influenced than
height of 2 feet—with one significant figure—is the least precise. the mean to extreme values, called outliers. An outlier is a data
Note that if you first round 4.33 to 4 and then do the calculation, point that is very different from all of the other data points and

you get an answer of 4 × 2 = 8 which is incorrect. therefore one that falls outside the overall pattern of a group of

,
20 UNIT 1 CHEMISTRY OF LIFE
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