The 2.5x thing I've heard so much about

Question

I like stats. Here is what I think, am I right?:

"According to the most recent census data, there are nearly 160 million more white people in America than there are black people. White people make up roughly 62 percent of the U.S. population but only about 49 percent of those who are killed by police officers. African Americans, however, account for 24 percent of those fatally shot and killed by the police despite being just 13 percent of the U.S. population. As The Post noted in a new analysis published last week, that means black Americans are 2.5 times as likely as white Americans to be shot and killed by police officers."

- https://www.chicagotribune.com/nation-world/ct-police-shootings-race-20160711-story.html

Here is some math for killings/population%:
White American: 49%/62% = .79
African American: 24%/13% = 1.85

Multiply .79 by 2.5 to "check the math"
.79 * 2.5 = 1.975
This is only 0.125‬ off from the African American number. Is this the math that proves African Americans are 2.5x more likely to be fatally shot by police than White Americans?

From what I know about sample size and populations, you do not take statistics from an entire population if that entire population is not involved. Here we compare the % of the USA of African Americans and White Americans in a case about police killings. Would it not be correct to compare all police involved with the two populations and then of those two populations, find the % that led to fatality cases via gunfire?

Answer 1

Without using actual figures we can look at the problem algebraically.

If P=total population of the States, W=population of “whites”, B=population of “blacks”, O=population of “others”, including indigenous Americans, P=W+B+O. If D(W) represents the number of deaths of whites in police custody and D(B) that of blacks under the same circumstances (excludes those whose deaths were not caused directly by one or more law enforcement officers), then the proportion of such deaths relative to the populations of the two groups would be D(W)/W and D(B)/B. If r is the multiplication factor such that D(B)/B=rD(W)/W and W-B=160m (160 million) then we can replace W with B+160m:

D(B)/B=rD(W)/(B+160m).

Total such deaths=D(B)+D(W). D(W)=0.49(D(B)+D(W)), D(B)=0.24(D(B)+D(W)).

Therefore, 0.24(D(B)+D(W))/B=0.49r(D(B)+D(W))/(B+160m), 0.24/B=0.49r/(B+160m).

0.24(B+160m)=0.49rB, 38.4m=B(0.49r-0.24).

To find r we need to know B. P=329m approx, and B=0.13P=42.8m approx.

0.49r-0.24=38.4/42.8, r=(38.4/42.8+0.24)/0.49=2.32 approx.