Question

if a new data point at 12 is added to the graph, which will be true? the mean will increase, and the median will stay the same. the median will increase, and the mean will stay the same. the mean will increase more than the median, but both will increase. the median will increase more than the mean, but both will increase.

Explanation:

Step1: Count original data points

From the dot plot: 1 (1), 2 (3), 3 (1), 4 (3), 5 (1). Total points: \(1 + 3 + 1 + 3 + 1 = 9\). Data set: \([1, 2, 2, 2, 3, 4, 4, 4, 5]\).

Step2: Find original mean and median

- Median (middle of 9 terms, 5th term): \(3\).
- Mean: \(\frac{1 + 2+2+2+3+4+4+4+5}{9}=\frac{27}{9} = 3\).

Step3: Add new data point (12)

New data set: \([1, 2, 2, 2, 3, 4, 4, 4, 5, 12]\) (10 terms).

Step4: Find new mean and median

- Median (average of 5th and 6th terms): \(\frac{3 + 4}{2}=3.5\) (wait, no—wait, 10 terms: positions 5 and 6. Wait, original data: 9 terms, sorted. After adding 12, sorted: \(1,2,2,2,3,4,4,4,5,12\). 5th term: 3, 6th term: 4. Median: \(\frac{3 + 4}{2}=3.5\)? Wait, no, wait original data: let's re - check. Original data: 1 (1), 2 (3), 3 (1), 4 (3), 5 (1). So sorted: 1, 2, 2, 2, 3, 4, 4, 4, 5. Median is 5th term: 3. After adding 12, sorted: 1, 2, 2, 2, 3, 4, 4, 4, 5, 12. Now, 10 terms, median is average of 5th (\(3\)) and 6th (\(4\)): \(\frac{3 + 4}{2}=3.5\). Wait, but wait, maybe I made a mistake earlier. Wait, no—wait, original number of data points: 9 (odd), median is 5th term (3). After adding 12, number of data points: 10 (even), median is average of 5th and 6th terms. 5th term: 3, 6th term: 4. So median becomes \(3.5\). Wait, but the mean: original mean 3, new mean: \(\frac{27+12}{10}=\frac{39}{10}=3.9\). Wait, but the options: let's re - evaluate. Wait, maybe my initial calculation was wrong. Wait, original data: 1 (1), 2 (3), 3 (1), 4 (3), 5 (1). So sum: \(1\times1 + 2\times3+3\times1 + 4\times3+5\times1=1 + 6+3 + 12+5 = 27\). Mean: \(27/9 = 3\). After adding 12, sum is \(27 + 12=39\), number of terms is 10, mean is \(39/10 = 3.9\) (increase). Median: original median (9 terms) is 5th term: 3. New median (10 terms) is average of 5th (3) and 6th (4) terms: \((3 + 4)/2=3.5\) (increase). Wait, but the options: the first option says median stays same, which is wrong. Wait, maybe I misread the dot plot. Wait, let's re - check the dot plot: positions 1:1 dot, 2:3 dots, 3:1 dot, 4:3 dots, 5:1 dot. So data points: 1, 2, 2, 2, 3, 4, 4, 4, 5. Correct. After adding 12, data points: 1, 2, 2, 2, 3, 4, 4, 4, 5, 12. Now, median: for 10 data points, the median is the average of the 5th and 6th values. 5th value: 3, 6th value: 4. So median is \((3 + 4)/2 = 3.5\) (increase). Mean: from 3 to \(39/10 = 3.9\) (increase). But the options: wait, the first option says mean increases, median stays same. That's wrong. Wait, maybe I made a mistake in the dot plot. Wait, maybe the dot plot is: 1:1, 2:3, 3:1, 4:3, 5:1. Wait, maybe the original median is 3, and after adding 12, the median is still 3? No, because with 10 data points, it's the average of 5th and 6th. Wait, no—wait, maybe the dot plot is different. Wait, maybe the original data has 9 points, and when we add 12, the number of points is 10. Wait, maybe I messed up the median calculation. Wait, let's list all data points:

Original: 1, 2, 2, 2, 3, 4, 4, 4, 5 (9 points).

After adding 12: 1, 2, 2, 2, 3, 4, 4, 4, 5, 12 (10 points).

Now, let's find the median:

For 9 points, median is the 5th term: 3.

For 10 points, median is the average of the 5th and 6th terms. The 5th term is 3, the 6th term is 4. So median is \((3 + 4)/2=3.5\). So median increases from 3 to 3.5.

Mean: original mean is \(27/9 = 3\), new mean is \((27 + 12)/10=39/10 = 3.9\). So mean increases from 3 to 3.9, median increases from 3 to 3.5. Wait, but the options: the third option says "The mean will increase more than the median, but both will increase." Let's check the increase:

Mean increase: \(3.9−3 = 0.9\).

Median increase: \(3.5−3 = 0.5\).

So mean increases more than median, and both increase. Wait, but earlier I thought the first option was wrong. Let's re - check the options:

Option 1: Mean increase, median same. Wrong, median increases.

Option 2: Median increase, mean same. Wrong, mean increases.

Option 3: Mean increase more than median, both increase. Correct, as mean increase (0.9) > median increase (0.5).

Option 4: Median increase more than mean. Wrong.

Wait, maybe my initial median calculation was wrong. Wait, no—wait, let's re - sort the data after adding 12:

1, 2, 2, 2, 3, 4, 4, 4, 5, 12.

Positions:

1:1

2:2

3:2

4:2

5:3

6:4

7:4

8:4

9:5

10:12

So 5th term: 3, 6th term: 4. Median: (3 + 4)/2 = 3.5. Original median: 3. So median increases by 0.5. Mean increases by 0.9. So mean increases more than median, and both increase.

Answer:

The mean will increase more than the median, but both will increase.