The total income for African American households making...

The total income for African American households making under $\$ 250,000$ in 2008 is given in the following table. Source: U.S. Census Bureau.
Use the table to estimate the mean income for African American households in 2008.

Key Concepts

Arithmetic Mean

The arithmetic mean is the sum of all values in a data set divided by the number of values. In the context of income data, it represents the average income per household. This concept is fundamental in statistics when describing the central tendency of a set of numerical data.

Grouped Data

Grouped data refers to data that has been organized into intervals or classes rather than listed individually. When the data is grouped, the exact values are not known, and instead, each group is approximated by a representative value, often the midpoint of the interval, to make statistical estimates such as the mean.

Midpoint

The midpoint is the value that is halfway between the lower and upper boundaries of a class interval. For grouped data, the midpoint is used as an estimate for all the values in that interval. This approach simplifies calculations when estimating the mean or other descriptive statistics for the data.

Weighted Average

A weighted average is an average in which each value in the data set is multiplied by a weight that reflects its relative importance or frequency. In the estimation of a mean for grouped data, each class’s midpoint is multiplied by the frequency of the class, and the sum of these products is divided by the total frequency. This method provides a more accurate estimate than a simple average for data organized into groups.

Estimation Techniques

Estimation techniques in statistics involve using approximations in place of exact values to analyze data. When data is presented in grouped form, analysts use methods like taking midpoints and calculating weighted averages to estimate overall measures such as the mean, acknowledging that these results are approximations rather than exact calculations.

Transcript

00:01 This question shows us a frequency table for incomes of african -american families whose salaries are less than $250 ,000.

00:12 It wants us to find the mean salary, or at least estimate it using this frequency table.

00:18 We know when we're estimating means from frequency tables that x bar is equal to the sum of the value times the frequency divided by the sum of the frequency.

00:32 In this table, what is being counted is the ranges.

00:37 So we don't exactly have an x, but we are given the midpoint of each range, and so we'll use that as x.

00:45 So i'm going to write down a small part of this table, and we'll go from there.

00:51 I'm going to write the midpoint, but instead of writing it as just dollars, i'm going to write it as thousands of dollars.

00:57 It'll make things a little bit easier, simpler to read.

01:00 So the first entry is $10 ,000, then $30 ,000, 50 ,000, 70 ,000, 90 ,000, 125 ,000, and 225 ,000.

01:16 Then we have frequencies, we have, and this is already in thousands.

01:22 Really, so this is really 4 ,378 ,3 ,8 ,000, 2 ,000 ,000, $891 ,500 ,000, 2 ,500 ,000, 1 ,000 ,000.

01:35 497 ,86 ,986 ,998, 398, 309, and 79 ,000.

01:46 Now i will make a column for x times f.

01:49 This i'm just going to do in a calculator.

01:51 I'll multiply my x column times my f column.

01:57 43 ,780, 1167730, 125 ,000.

02:10 104 ,790, 77 ,940, 124 ,750, 125, 540, or 54 ,000, 75, and finally 17 ,775...

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