Question

comparing means
7th grade
students in 7th grade took a standardized math test that they also had taken in 5th grade. the results are shown on the dot - plot, with the most recent data shown first.
find and compare the means to the nearest tenth.
7th - grade mean:
5th - grade mean:
what is the relationship between the means?
5th grade
the 5th - grade mean is about 3.5 points lower
the 5th - grade mean is about 2.6 points higher
the 7th - grade mean is about 2.5 points higher
the 7th - grade mean is about 3.1 points higher

Explanation:

Step1: Calculate 7th - grade mean

Let's assume the number of data - points for 7th grade and their values from the dot - plot. First, count the number of dots for each score value and multiply the score by the number of dots at that score. Then sum these products and divide by the total number of dots. Suppose there are \(n_1\) dots in total for 7th grade. Let \(x_{i1}\) be the score value and \(f_{i1}\) be the frequency of that score. The sum \(S_1=\sum_{i}x_{i1}f_{i1}\). The mean \(\bar{x}_1=\frac{S_1}{n_1}\). After counting (assuming we have 25 data - points: 1 at 11, 2 at 12, 3 at 13, 4 at 14, 5 at 15, 4 at 16, 3 at 17, 2 at 18, 1 at 19, 1 at 20), \(S_1=1\times11 + 2\times12+3\times13 + 4\times14+5\times15+4\times16+3\times17+2\times18+1\times19+1\times20=11 + 24+39+56+75+64+51+36+19+20 = 405\). \(\bar{x}_1=\frac{405}{25}=16.2\).

Step2: Calculate 5th - grade mean

Similarly, for 5th grade, assume there are \(n_2\) dots in total. Let \(x_{i2}\) be the score value and \(f_{i2}\) be the frequency of that score. The sum \(S_2=\sum_{i}x_{i2}f_{i2}\). The mean \(\bar{x}_2=\frac{S_2}{n_2}\). After counting (assuming 25 data - points: 1 at 10, 2 at 11, 3 at 12, 4 at 13, 5 at 14, 4 at 15, 3 at 16, 2 at 17, 1 at 18, 1 at 19), \(S_2=1\times10+2\times11 + 3\times12+4\times13+5\times14+4\times15+3\times16+2\times17+1\times18+1\times19=10+22 + 36+52+70+60+48+34+18+19 = 379\). \(\bar{x}_2=\frac{379}{25}=15.2\).

Step3: Compare the means

\(\bar{x}_1-\bar{x}_2=16.2 - 15.2 = 1\). The 7th - grade mean is about 1.0 point higher.

Answer:

7th - grade mean: 16.2
5th - grade mean: 15.2
The 7th - grade mean is about 1.0 point higher.