examples

Marci’s exam scores for her last math class were 79, 86, 82, and 94. What would the mean of these values would be?

Answer: [latex]\frac{79+86+82+94}{4}=85.25[/latex]. Typically we round means to one more decimal place than the original data had. In this case, we would round 85.25 to 85.3.

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The number of touchdown (TD) passes thrown by each of the 31 teams in the National Football League in the 2000 season are shown below. 37 33 33 32 29 28 28 23 22 22 22 21 21 21 20 20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6 What is the mean number of TD passes?

Answer: Adding these values, we get 634 total TDs. Dividing by 31, the number of data values, we get 634/31 = 20.4516. It would be appropriate to round this to 20.5. It would be most correct for us to report that “The mean number of touchdown passes thrown in the NFL in the 2000 season was 20.5 passes,” but it is not uncommon to see the more casual word “average” used in place of “mean.”

Both examples are described further in the following video. https://youtu.be/3if9Le2sO0c

examples

The one hundred families in a particular neighborhood are asked their annual household income, to the nearest $5 thousand dollars. The results are summarized in a frequency table below.

Income (thousands of dollars)	Frequency
15	6
20	8
25	11
30	17
35	19
40	20
45	12
50	7

What is the mean average income in this neighborhood?

Answer: Calculating the mean by hand could get tricky if we try to type in all 100 values: [latex-display]\frac{\overbrace{15+\cdots+15}^{\text{6terms}}+\overbrace{20+\cdots+20}^{\text{8terms}}+\overbrace{25+\cdots+25}^{\text{11terms}}+\cdots}{\text{100}}[/latex-display] We could calculate this more easily by noticing that adding 15 to itself six times is the same as = 90. Using this simplification, we get [latex-display]\frac{15\cdot6+20\cdot8+25\cdot11+30\cdot17+35\cdot19+40\cdot20+45\cdot12+50\cdot7}{\text{100}}=\frac{3390}{100}=33.9[/latex-display] The mean household income of our sample is 33.9 thousand dollars ($33,900).

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Extending off the last example, suppose a new family moves into the neighborhood example that has a household income of $5 million ($5000 thousand). What is the new mean of this neighborhood's income?

Answer: Adding this to our sample, our mean is now: [latex-display]\frac{15\cdot6+20\cdot8+25\cdot11+30\cdot17+35\cdot19+40\cdot20+45\cdot12+50\cdot7+5000\cdot1}{\text{101}}=\frac{8390}{101}=83.069[/latex-display]

Both situations are explained further in this video. https://youtu.be/1_4Hxcq8DpQ

example

Returning to the football touchdown data, we would start by listing the data in order. Luckily, it was already in decreasing order, so we can work with it without needing to reorder it first. 37 33 33 32 29 28 28 23 22 22 22 21 21 21 20 20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6 What is the median TD value?

Answer: Since there are 31 data values, an odd number, the median will be the middle number, the 16th data value (31/2 = 15.5, round up to 16, leaving 15 values below and 15 above). The 16th data value is 20, so the median number of touchdown passes in the 2000 season was 20 passes. Notice that for this data, the median is fairly close to the mean we calculated earlier, 20.5.

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Find the median of these quiz scores: 5 10 8 6 4 8 2 5 7 7

Answer: We start by listing the data in order: 2 4 5 5 6 7 7 8 8 10 Since there are 10 data values, an even number, there is no one middle number. So we find the mean of the two middle numbers, 6 and 7, and get (6+7)/2 = 6.5. The median quiz score was 6.5.

Learn more about these median examples in this video. https://youtu.be/WEdr_rSRObk

Example

Let us return now to our original household income data

Income (thousands of dollars)	Frequency
15	6
20	8
25	11
30	17
35	19
40	20
45	12
50	7

What is the mean of this neighborhood's household income?

Answer: Here we have 100 data values. If we didn’t already know that, we could find it by adding the frequencies. Since 100 is an even number, we need to find the mean of the middle two data values - the 50th and 51st data values. To find these, we start counting up from the bottom: There are 6 data values of $15, so                  Values 1 to 6 are $15 thousand The next 8 data values are $20, so                  Values 7 to (6+8)=14 are $20 thousand The next 11 data values are $25, so                Values 15 to (14+11)=25 are $25 thousand The next 17 data values are $30, so                Values 26 to (25+17)=42 are $30 thousand The next 19 data values are $35, so                Values 43 to (42+19)=61 are $35 thousand From this we can tell that values 50 and 51 will be $35 thousand, and the mean of these two values is $35 thousand. The median income in this neighborhood is $35 thousand.

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If we add in the new neighbor with a $5 million household income, then there will be 101 data values, and the 51st value will be the median. As we discovered in the last example, the 51st value is $35 thousand. Notice that the new neighbor did not affect the median in this case. The median is not swayed as much by outliers as the mean is. View more about the median of this neighborhood's household incomes here. https://youtu.be/kqEu9EDkmfU

Example

In our vehicle color survey earlier in this section, we collected the data

Color	Frequency
Blue	3
Green	5
Red	4
White	3
Black	2
Grey	3

Which color is the mode?

Answer: For this data, Green is the mode, since it is the data value that occurred the most frequently.

Mode in this example is explained by the video here. https://youtu.be/pFpkWrib3Jk