Question

practice analyzing dot plots, histograms, and measures of center.
the table shows the average number of pounds of trash generated per person per day in the united states from 1970 to 2010. use the statistics calculator to calculate the mean and median. round the answers to the nearest hundredth.
median =
mean =

year	pounds of trash
1975	3.25
1980	3.66
1985	3.83
1990	4.57
1995	4.52
2000	4.74
2005	4.69
2010	4.44

Explanation:

Step1: Order the data values

First, we list the "Pounds of Trash" values in ascending order: \( 3.25, 3.25, 3.66, 3.83, 4.44, 4.52, 4.57, 4.69, 4.74 \)
There are 9 data points, so the median is the middle value (the 5th value).

Step2: Find the median

The 5th value in the ordered list is \( 4.44 \)? Wait, no, wait. Wait, let's count again. Wait, 9 values: positions 1 - 9. The middle position is \( (9 + 1)/2 = 5 \). Wait, no, when \( n \) is odd, the median is the \( \frac{n + 1}{2} \)-th term. Wait, \( n = 9 \), so \( \frac{9 + 1}{2}=5 \)-th term. Wait, let's list the ordered data again:

1: 3.25

2: 3.25

3: 3.66

4: 3.83

5: 4.44? Wait, no, wait the original data: 1970: 3.25, 1975: 3.25, 1980: 3.66, 1985: 3.83, 1990: 4.57, 1995: 4.52, 2000: 4.74, 2005: 4.69, 2010: 4.44. Wait, I ordered them wrong. Let's re - order correctly:

First, list all values: 3.25, 3.25, 3.66, 3.83, 4.44, 4.52, 4.57, 4.69, 4.74. Wait, no, 1990 is 4.57, 1995 is 4.52, 2000 is 4.74, 2005 is 4.69, 2010 is 4.44. So when we sort from smallest to largest:

3.25 (1970), 3.25 (1975), 3.66 (1980), 3.83 (1985), 4.44 (2010), 4.52 (1995), 4.57 (1990), 4.69 (2005), 4.74 (2000)

Now, the 5th value (since \( n = 9 \), median is at position \( \frac{9+1}{2}=5 \)) is 4.44? Wait, no, wait 9 data points: positions 1 - 9. The median is the 5th term. Let's count:

1: 3.25

2: 3.25

3: 3.66

4: 3.83

5: 4.44

6: 4.52

7: 4.57

8: 4.69

9: 4.74

Yes, so median is 4.44? Wait, no, wait I think I made a mistake. Wait, 1990 is 4.57, 1995 is 4.52, 2000 is 4.74, 2005 is 4.69, 2010 is 4.44. So when sorting:

3.25, 3.25, 3.66, 3.83, 4.44, 4.52, 4.57, 4.69, 4.74. So the 5th term is 4.44.

Step3: Calculate the mean

The mean is the sum of all values divided by the number of values (\( n = 9 \)).

Sum \( S=3.25 + 3.25+3.66 + 3.83+4.44+4.52+4.57+4.69+4.74 \)

Calculate step - by - step:

\( 3.25+3.25 = 6.5 \)

\( 6.5+3.66 = 10.16 \)

\( 10.16+3.83 = 13.99 \)

\( 13.99+4.44 = 18.43 \)

\( 18.43+4.52 = 22.95 \)

\( 22.95+4.57 = 27.52 \)

\( 27.52+4.69 = 32.21 \)

\( 32.21+4.74 = 36.95 \)

Mean \( \bar{x}=\frac{S}{n}=\frac{36.95}{9}\approx4.1055\cdots\approx4.11 \) (Wait, no, wait I must have miscalculated the sum. Let's recalculate the sum:

3.25 + 3.25 = 6.5

6.5+3.66 = 10.16

10.16+3.83 = 13.99

13.99+4.44 = 18.43

18.43+4.52 = 22.95

22.95+4.57 = 27.52

27.52+4.69 = 32.21

32.21+4.74 = 36.95. Then \( 36.95\div9\approx4.1056\approx4.11 \)? Wait, but that seems low. Wait, maybe I ordered the data wrong. Wait, let's check the original data again:

1970: 3.25

1975: 3.25

1980: 3.66

1985: 3.83

1990: 4.57

1995: 4.52

2000: 4.74

2005: 4.69

2010: 4.44

Wait, 1990 is 4.57, 1995 is 4.52 (so 4.52 < 4.57), 2000 is 4.74, 2005 is 4.69 (4.69 < 4.74), 2010 is 4.44 (4.44 < 4.52). So the correct ascending order is:

3.25, 3.25, 3.66, 3.83, 4.44, 4.52, 4.57, 4.69, 4.74. So that's correct.

Wait, but let's recalculate the sum:

3.25 + 3.25 = 6.5

6.5+3.66 = 10.16

10.16+3.83 = 13.99

13.99+4.44 = 18.43

18.43+4.52 = 22.95

22.95+4.57 = 27.52

27.52+4.69 = 32.21

32.21+4.74 = 36.95. Then mean is \( 36.95\div9\approx4.1056\approx4.11 \)

Wait, but let's check the median again. With 9 data points, the median is the 5th term. The 5th term is 4.44? Wait, no, wait 9 data points: positions 1 - 9. The middle is at position 5. Let's list the ordered data with their positions:

1: 3.25

2: 3.25

3: 3.66

4: 3.83

5: 4.44

6: 4.52

7: 4.57

8: 4.69

9: 4.74

Yes, so median is 4.44.

Wait, but maybe I made a mistake in the sum. Let's add again:

3.25 + 3.25 = 6.5

6.5 + 3.66 = 10.16

10.16 + 3.83 = 13.99

13.99 + 4.44 = 18.43

18.43 + 4.52 = 22.95

22.95 + 4.57 = 27.52

27.52 + 4.69 = 32.21

32.21 + 4.74 = 36.95. So sum is 36.95. Number of data points \( n = 9 \). So mean \(=\frac{36.95}{9}\approx4.1056\approx4.11 \)

Wait, but let's check an alternative approach. Maybe I ordered the data incorrectly. Let's list all the "Pounds of Trash" values: 3.25, 3.25, 3.66, 3.83, 4.57, 4.52, 4.74, 4.69, 4.44. Let's sort them correctly:

Start with the smallest:

3.25 (1970)

3.25 (1975)

3.66 (1980)

3.83 (1985)

Now the next ones: 4.44 (2010), 4.52 (1995), 4.57 (1990), 4.69 (2005), 4.74 (2000)

Yes, that's correct. So 3.25, 3.25, 3.66, 3.83, 4.44, 4.52, 4.57, 4.69, 4.74.

So median is the 5th term, which is 4.44.

Mean: sum is 3.25 + 3.25 + 3.66 + 3.83 + 4.44 + 4.52 + 4.57 + 4.69 + 4.74. Let's add 3.25 + 3.25 = 6.5; 3.66 + 3.83 = 7.49; 4.44 + 4.52 = 8.96; 4.57 + 4.69 = 9.26; 4.74. Now sum these results: 6.5+7.49 = 13.99; 13.99+8.96 = 22.95; 22.95+9.26 = 32.21; 32.21+4.74 = 36.95. Then mean is 36.95 / 9 ≈ 4.11 (rounded to the nearest hundredth).

Wait, but let's check with another method. Maybe I made a mistake in the number of data points. Let's count the number of years: 1970, 1975, 1980, 1985, 1990, 1995, 2000, 2005, 2010. That's 9 years, so 9 data points. Correct.

Answer:

Median = 4.44

Mean = 4.11