{"id":33,"date":"2023-11-28T17:07:25","date_gmt":"2023-11-28T22:07:25","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/weighted-mean\/"},"modified":"2024-04-18T04:22:42","modified_gmt":"2024-04-18T08:22:42","slug":"weighted-mean","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/weighted-mean\/","title":{"raw":"Weighted Mean","rendered":"Weighted Mean"},"content":{"raw":"
\r\n
\r\n

Learning Outcomes<\/p>\r\n \r\n\r\n<\/header>\r\n

\r\n\r\nUnderstand the difference between a regular and weighted mean. Learn how to calculated a weighted mean and calculate for a missing weight or data value.\r\n\r\n<\/div>\r\n<\/div>\r\n\"\"\r\n\r\n\"The weighted arithmetic mean is similar to an ordinary\u00a0arithmetic mean<\/a>\u00a0(the most common type of\u00a0average<\/a>), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others.\"[footnote]https:\/\/en.wikipedia.org\/wiki\/Weighted_arithmetic_mean[\/footnote]\r\n\r\nThis kind of average is called the weighted mean, and is given by the following formula:\r\n\r\n\\[x_w =\\frac{\\Sigma x\\cdot w}{\\Sigma w}\\]\r\n\r\nWhere the x's are the data you are looking to average and the w's are the weights assigned to each of those data points.\r\n

Illustration<\/h1>\r\nSuppose you have $3,000, and make 3 different investments, some invested at 5%, some at 6%, and some at 7%.\r\n\r\nQuestion:<\/strong><\/span> Is your average rate of return equal to [latex]\\frac{(5 + 6 + 7)\\%} {3} = 6%[\/latex]?\r\n\r\nAnswer:<\/strong><\/span> The answer ONLY if you invested equal amounts in each investment, and NO otherwise. In the case where you invest different amounts, a weighted average is required to determine the average rate of return.\r\n

Example: Investing Different Amounts in Three Investments<\/h1>\r\n

Example 4.1.1<\/h2>\r\nLet us suppose you invested $30,000 at 5%, $6,000 at 6%, and $4,500 at 7%.\r\n\r\nQuestion:<\/strong><\/span> Is the overall (average) rate of return for your investment 6%?\r\n\r\nAnswer:<\/strong><\/span> No, it is given by the following calculation:\r\n

[latex]\r\n\\begin{align*}\r\nx_w &=\\frac{5\\%\\cdot$30,000 +6\\%\\cdot$6,000 + 7\\%\\cdot$4,500}{$30,000+$6,000+$4,500}\\\\\r\n&=\\frac{$1,500 +$360 + $315}{$30,000+$6,000+$4,500}\\\\\r\n&= \\frac{$2,175}{$40,500}\\\\\r\n&= 0.0537 = 5.37\\%\r\n\\end{align*}\r\n[\/latex]<\/p>\r\n\r\n

Example: Tables for Weighted Mean Calculations<\/h1>\r\n

Example 4.1.2<\/h2>\r\nLet us, instead, use a table to organize the calculations from the previous example. Note: the x<\/em> \u00b7<\/em> w<\/em> column is the product of the x <\/em>and w<\/em> values.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
x<\/em><\/th>\r\nw<\/em><\/th>\r\nx<\/em> \u00b7<\/em> w<\/em><\/th>\r\n<\/tr>\r\n
5%<\/td>\r\n$30,000<\/td>\r\n0.<\/em>05 \u00d7<\/em> $30,<\/em>000 = $1,<\/em>100<\/td>\r\n<\/tr>\r\n
6%<\/td>\r\n$6,000<\/td>\r\n0.<\/em>06 \u00d7<\/em> $6,<\/em>000 = $360<\/td>\r\n<\/tr>\r\n
7%<\/td>\r\n$4,500<\/td>\r\n0.<\/em>07 \u00d7<\/em> $4,<\/em>500 = $315<\/td>\r\n<\/tr>\r\n
Total<\/strong><\/td>\r\n$40,500<\/strong><\/td>\r\n$2,175<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis gives the following results:\r\n

[latex]\r\n\\begin{align*}\r\nx_w &=\\frac{\\Sigma x\\cdot w}{\\Sigma w} =\\frac{$40,500}{$2,175} =0.0537=5.37\\%\r\n\\end{align*}\r\n[\/latex]<\/p>\r\n\r\n

Example: Excel for Weighted Mean (Video)<\/h1>\r\n

Example 4.1.3<\/h2>\r\nWe can also use Excel's =sum()<\/a> and =sumproduct<\/a> formulas to calculate the weighted means:\r\n\r\n[embed]https:\/\/youtu.be\/UEdeP1d2-XA?si=qnxf_-3WR1zohpDn[\/embed]\r\n\r\nClick here for the Excel solutions shown in the above video<\/a>.\r\n

Example: Solving for Missing x-value (X)<\/h1>\r\n

Example 4.2.1<\/h2>\r\nProblem Setup:<\/strong><\/span> You purchased shares several years ago and sold them today:\r\n