1.2 Percents

In the 2004 vice-presidential debates, Edwards's claimed that US forces have suffered "90% of the coalition casualties" in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies "have taken almost 50 percent" of the casualties[1]. Who is correct? How can we make sense of these numbers?

Percent literally means “per 100,” or “parts per hundred.” When we write 40%, this is equivalent to the fraction $\frac{40}{100}$ or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since $\frac{80}{200} = \frac{10}{25} = \frac{40}{100}$ .

The base of a percent is very important. For example, while Nixon was president, it was argued that marijuana was a “gateway” drug, claiming that 80% of marijuana smokers went on to use harder drugs like cocaine. The problem is, this isn’t true. The true claim is that 80% of harder drug users first smoked marijuana. The difference is one of base: 80% of marijuana smokers using hard drugs, vs. 80% of hard drug users having smoked marijuana. These numbers are not equivalent. As it turns out, only one in 2,400 marijuana users actually go on to use harder drugs[2].

Example 6

There are about 75 QFC supermarkets in the U.S. Albertsons has about 215 stores. Compare the size of the two companies.

Show solution

When we make comparisons, we must ask first whether an absolute or relative comparison. The absolute difference is $215 – 75 equals 140$ . From this, we could say “Albertsons has 140 more stores than QFC.” However, if you wrote this in an article or paper, that number does not mean much. The relative difference may be more meaningful. There are two different relative changes we could calculate, depending on which store we use as the base:

Using QFC as the base, $\frac{140}{75} = 1.867$ .

This tells us Albertsons is 186.7% larger than QFC.

Using Albertsons as the base, $\frac{140}{215} = 0.651$ .

This tells us QFC is 65.1% smaller than Albertsons.

Notice both of these are showing percent differences. We could also calculate the size of Albertsons relative to QFC:, which tells us Albertsons is 2.867 times the size of QFC. Likewise, we could calculate the size of QFC relative to Albertsons:, which tells us that QFC is 34.9% of the size of Albertsons.

Example 7

Suppose a stock drops in value by 60% one week, then increases in value the next week by 75%. Is the value higher or lower than where it started?

# Percent change explorer: every step is measured against a NEW base.
# CHANGE THESE two lines and re-run.
start   = 100.00
changes = [-60, +75]      # the book's stock: down 60%, then up 75%
# Things worth trying:
#   [-50, +50]  a 50% fall then a 50% rise does NOT get you back to the start
#   [+50, -50]  the same ending value - the ORDER does not matter, the BASE does
#   [-10, -10, -10]  three 10% cuts is a 27.1% cut, not a 30% cut

value = start
print("step   % change        base   abs change    new value")
print("-" * 55)
for i, pct in enumerate(changes, 1):
    base  = value                 # <-- the base is the CURRENT value, not the original
    delta = base * pct / 100
    value = base + delta
    print("%-6d %+8.1f%% %11.2f %12.2f %12.2f" % (i, pct, base, delta, value))
print("-" * 55)

absolute = abs(value - start)
relative = (value - start) / start * 100     # base = the ORIGINAL value
print("start %.2f  ->  end %.2f" % (start, value))
print("absolute change : %.2f" % absolute)
print("relative change : %+.1f%%   (base = the original %.2f)" % (relative, start))

naive = sum(changes)
print("just adding the percents gives %+.1f%% -- wrong by %+.1f percentage points"
      % (naive, naive - relative))
print()
print("Why: %+.1f%% of %.2f is not the same amount of money as %+.1f%% of %.2f."
      % (changes[0], start, changes[-1], start))

Show solution

To answer this question, suppose the value started at $100. After one week, the value dropped by 60%:

$$ 100 - $ 100 (0.60) = $ 100 - $ 60 = $ 40$ .

In the next week, notice that base of the percent has changed to the new value, $40. Computing the 75% increase:

$$ 40 + $ 40 (0.75) = $ 40 + $ 30 = $ 70$ .

In the end, the stock is still $30 lower, or $\frac{$ 30}{$ 100} = 30 %$ lower, valued than it started.

Example 8

A Seattle Times article on high school graduation rates reported “The number of schools graduating 60 percent or fewer students in four years – sometimes referred to as “dropout factories” – decreased by 17 during that time period. The number of kids attending schools with such low graduation rates was cut in half.”

a) Is the “decrease by 17” number a useful comparison?

b) Considering the last sentence, can we conclude that the number of “dropout factories” was originally 34?

Show solution

a) This number is hard to evaluate, since we have no basis for judging whether this is a larger or small change. If the number of “dropout factories” dropped from 20 to 3, that’d be a very significant change, but if the number dropped from 217 to 200, that’d be less of an improvement.

b) The last sentence provides relative change which helps put the first sentence in perspective. We can estimate that the number of “dropout factories” was probably previously around 34. However, it’s possible that students simply moved schools rather than the school improving, so that estimate might not be fully accurate.

Example 9

Show solution

Without more information, it is hard for us to judge who is correct, but we can easily conclude that these two percents are talking about different things, so one does not necessarily contradict the other. Edward’s claim was a percent with coalition forces as the base of the percent, while Cheney’s claim was a percent with both coalition and Iraqi security forces as the base of the percent. It turns out both statistics are in fact fairly accurate.

Your Turn

Try it Now 3

In the 2012 presidential elections, one candidate argued that “the president’s plan will cut $716 billion from Medicare, leading to fewer services for seniors,” while the other candidate rebuts that “our plan does not cut current spending and actually expands benefits for seniors, while implementing cost saving measures.” Are these claims in conflict, in agreement, or not comparable because they’re talking about different things?

Answer

Without more information, it is hard to judge these arguments. This is compounded by the complexity of Medicare. As it turns out, the $716 billion is not a cut in current spending, but a cut in future increases in spending, largely reducing future growth in health care payments. In this case, at least the numerical claims in both statements could be considered at least partially true. Here is one source of more information if you’re interested: http://factcheck.org/2012/08/a-campaign-full-of-mediscare/

We’ll wrap up our review of percents with a couple cautions. First, when talking about a change of quantities that are already measured in percents, we have to be careful in how we describe the change.

Example 10

A politician’s support increases from 40% of voters to 50% of voters. Describe the change.

# "Up 10 points" and "up 25%" describe the SAME change. Both are honest;
# one of them always sounds bigger. Which one depends on the base.
# CHANGE this list of (before, after) poll percentages and re-run.
polls = [
    ("the book's politician", 40.0, 50.0),
    ("a fringe candidate",     2.0,  3.0),
    ("an incumbent",          80.0, 84.0),
    ("a tax that lost ground",44.0, 33.0),
]

print("%-24s %7s %7s %10s %12s" % ("case", "before", "after", "points", "percent chg"))
print("-" * 64)
for name, before, after in polls:
    points   = after - before                      # absolute change, in PERCENTAGE POINTS
    rel      = (after - before) / before * 100     # relative change, base = before
    print("%-24s %6.1f%% %6.1f%% %+9.1f %+11.1f%%" % (name, before, after, points, rel))
print("-" * 64)
print("Same 'points' column, wildly different 'percent chg' column: the base is doing the work.")
print()

# The reverse trap: a headline percent, back-converted to points.
HEADLINE_PERCENT_INCREASE = 50.0   # <-- CHANGE ME ("support surged 50%!")
BASE_SUPPORT              = 4.0    # <-- CHANGE ME (percent of voters, before)
after = BASE_SUPPORT * (1 + HEADLINE_PERCENT_INCREASE / 100)
print("'Support surged %.0f%%' starting from %.1f%% of voters" % (HEADLINE_PERCENT_INCREASE, BASE_SUPPORT))
print("  really means %.1f%% -> %.1f%%, a move of %.1f percentage points."
      % (BASE_SUPPORT, after, after - BASE_SUPPORT))

Show solution

We could describe this using an absolute change: $| 50 % - 40 % | = 10 %$ . Notice that since the original quantities were percents, this change also has the units of percent. In this case, it is best to describe this as an increase of 10 percentage points.

In contrast, we could compute the percent change: $\frac{10 %}{40 %} = 0.25 = 25 %$ increase. This is the relative change, and we’d say the politician’s support has increased by 25%.

Lastly, a caution against averaging percents.

Example 11

A basketball player scores on 40% of 2-point field goal attempts, and on 30% of 3-point of field goal attempts. Find the player’s overall field goal percentage.

# You cannot average percents unless the groups are the same size.
# CHANGE the attempt counts and watch the true overall percentage move
# while the (wrong) simple average never budges.
two_pt_rate, three_pt_rate = 0.40, 0.30     # the book's player

splits = [(200, 100),    # the book's example
          (100, 100),    # equal attempts -- ONLY here does the simple average work
          (280,  20),    # a post player
          ( 20, 280)]    # a three-point specialist

print("%8s %8s %10s %12s %12s" % ("2pt att", "3pt att", "made", "true FG%", "simple avg"))
print("-" * 54)
for a2, a3 in splits:
    made  = a2 * two_pt_rate + a3 * three_pt_rate
    true  = made / (a2 + a3) * 100
    naive = (two_pt_rate + three_pt_rate) / 2 * 100
    print("%8d %8d %10.0f %11.1f%% %11.1f%%" % (a2, a3, made, true, naive))
print("-" * 54)
print("The simple average is a WEIGHTED average that silently assumes equal weights.")
print()

# The general rule, written out:
a2, a3 = 200, 100                                   # <-- CHANGE ME
w2, w3 = a2 / (a2 + a3), a3 / (a2 + a3)
print("weighted average = %.2f*%.0f%% + %.2f*%.0f%% = %.1f%%"
      % (w2, two_pt_rate * 100, w3, three_pt_rate * 100,
         (w2 * two_pt_rate + w3 * three_pt_rate) * 100))
print("With %d and %d attempts the weights are %.2f and %.2f, not 0.50 and 0.50."
      % (a2, a3, w2, w3))

Show solution

It is very tempting to average these values, and claim the overall average is 35%, but this is likely not correct, since most players make many more 2-point attempts than 3-point attempts. We don’t actually have enough information to answer the question. Suppose the player attempted 200 2-point field goals and 100 3-point field goals. Then they made 200(0.40) = 80 2-point shots and $100 open parenthesis 0.30 close parenthesis equals 30$ 3-point shots. Overall, they made 110 shots out of 300, for a $\frac{110}{300} = 0.367 = 36.7 %$ overall field goal percentage.

[1] www.factcheck.org/cheney_edwards_mangle_facts.html

[2] http://tvtropes.org/pmwiki/pmwiki.php/Main/LiesDamnedLiesAndStatistics

[3] www.whitehouse.gov/sites/defa...s/hist07z1.xls

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.