Maria has the following scores on exams in her social studies class: 86, 75, 97, 58, 94, and 58.
Find the mean, median, and mode of the scores.
Should Maria’s social studies teacher use the mean, median, or mode of the exam scores to convince Maria that she needs to study more regularly for her exams?
Mean:
To find the mean, you add up all the scores and divide by the number of scores.
86 + 75 + 97 + 58 + 94 + 58 = 468
468 ÷ 6 = 78
The mean is 78.
Median:
To find the median, you need to put the scores in order from lowest to highest.
58, 58, 75, 86, 94, 97
There are six scores, so the median is the middle score, which is 86.
Mode:
The mode is the score that appears most frequently. In this case, there are two scores that appear twice (58 and 94) and the rest appear only once. So, there is no mode.
Based on these results, Maria's teacher should use the mean of the exam scores to convince her to study more regularly. The mean is lower than what Maria might expect, as she might assume her highest and lowest grades would balance out. By showing her the mean, Maria's teacher can illustrate the importance of consistent studying habits, because even one or two low scores can drag down an average.
The stem-and-leaf plot shows the heights in centimeters of Teddy Bear sunflowers grown in two different types of soil.
Soil A Soil B
5 9
5 2 1 1 6 3 9
5 1 0 7 0 2 3 6 7 8
2 1 8 3
0 9
Key: 9|6 means 69 Key: 5|8 means 58
Calculate the mean of each data set.
Calculate the mean absolute deviation (MAD) of each data set.
Which set is more variable? How do you know?
Mean and MAD of Soil A:
To find the mean of Soil A, we need to add up all the values and divide by the total number of values:
(10 + 11 + 12 + 15 + 21 + 51 + 52 + 59) / 8 = 33.125 cm
The mean height of Teddy Bear sunflowers grown in Soil A is 33.125 cm.
To find the MAD, we first need to find the deviation of each value from the mean. We can do this by subtracting the mean from each value:
10 - 33.125 = -23.125
11 - 33.125 = -22.125
12 - 33.125 = -21.125
15 - 33.125 = -18.125
21 - 33.125 = -12.125
51 - 33.125 = 17.875
52 - 33.125 = 18.875
59 - 33.125 = 25.875
Next, we take the absolute value of each deviation:
|-23.125| = 23.125
|-22.125| = 22.125
|-21.125| = 21.125
|-18.125| = 18.125
|-12.125| = 12.125
|17.875| = 17.875
|18.875| = 18.875
|25.875| = 25.875
Now we find the mean of these absolute deviations:
(23.125 + 22.125 + 21.125 + 18.125 + 12.125 + 17.875 + 18.875 + 25.875) / 8 = 19.5 cm
The MAD for Soil A is 19.5 cm.
Mean and MAD of Soil B:
To find the mean of Soil B, we need to add up all the values and divide by the total number of values:
(30 + 38 + 39 + 62 + 63 + 68 + 70 + 72 + 73 + 78 + 89) / 11 = 60.727 cm
The mean height of Teddy Bear sunflowers grown in Soil B is 60.727 cm.
To find the MAD, we follow the same steps as before:
|-29.727| = 29.727
|-22.727| = 22.727
|-21.727| = 21.727
|-6.727| = 6.727
|-3.727| = 3.727
|7.273| = 7.273
|8.273| = 8.273
|11.273| = 11.273
|12.273| = 12.273
|17.273| = 17.273
|28.273| = 28.273
(29.727 + 22.727 + 21.727 + 6.727 + 3.727 + 7.273 + 8.273 + 11.273 + 12.273 + 17.273 + 28.273) / 11 = 16.018 cm
The MAD for Soil B is 16.018 cm.
Which set is more variable?
To determine which set is more variable, we compare the MADs. The set with the higher MAD is generally considered more variable. In this case, Soil A has a MAD of 19.5 cm, while Soil B has a MAD of 16.018 cm. Therefore, Soil A is more variable. This makes sense because Soil A's data is much more spread out and has two very large deviations from the mean, while Soil B's data is more clustered around the mean.
To find the mean of the scores, you need to add up all the scores and divide the sum by the total number of scores. Let's calculate it step by step:
The given scores are: 86, 75, 97, 58, 94, and 58.
1. Add up all the scores: 86 + 75 + 97 + 58 + 94 + 58 = 468.
2. Divide the sum by the total number of scores (in this case, 6): 468 ÷ 6 = 78.
So, the mean score is 78.
To find the median of the scores, you need to arrange them in ascending order (from lowest to highest) and find the middle value. In this case, there are 6 scores, so the middle value will be between the 3rd and 4th scores. Let's calculate it step by step:
Arranging the scores in ascending order: 58, 58, 75, 86, 94, 97.
The middle two numbers are 75 and 86. So, to find the median, you need to take the average of these two numbers:
(75 + 86) ÷ 2 = 161 ÷ 2 = 80.5.
So, the median score is 80.5.
To find the mode of the scores, you need to identify the score that appears most frequently. In this case, the score 58 appears twice, which is more than any other number.
So, the mode of the scores is 58.
Now, let's consider which measure of central tendency Maria's social studies teacher should use to convince her to study more regularly.
Since the mean is impacted by extreme scores (like the low score of 58), it might not provide an accurate representation of Maria's overall performance. The median, on the other hand, is not affected by extreme values and gives a more balanced view of the distribution. Therefore, the teacher should use the median score to convince Maria that she needs to study more regularly for her exams.