Question

distance from school in blocks wynn needs to find the center of the data set shown on the dot plot. the dot plot has dots. the dot plot has dots to the left of the center and dots to the right of the center. the center of the data set is at 2 between 2 and 3 at 3 between 3 and 4

Explanation:

Step1: Count the total number of dots

Count each dot in the dot - plot. There are \(6 + 4+3 + 1+2 + 1=17\) dots.

Step2: Find the position of the center

For a data set with \(n = 17\) data points (dots), the median (center) is the \(\frac{n + 1}{2}=\frac{17+1}{2}=9^{th}\) data point when the data is ordered.

• Count the cumulative number of dots:
• For \(x = 1\), there are \(6\) dots.
• For \(x=2\), there are \(6 + 4=10\) dots.
• Since \(6\lt9\lt10\), the \(9^{th}\) dot is at \(x = 2\). But wait, we made a mistake above. Let's re - count:
• The total number of dots: \(6+4 + 3+1+2+1=17\). The median is the \(\frac{17 + 1}{2}=9^{th}\) value.
• Cumulative count:
• \(1\) block: \(6\) dots.
• \(2\) blocks: \(6 + 4=10\) dots.
• The \(9^{th}\) dot is within the group of \(2\) - block data. But actually, we should consider the balance.
• Let's use another approach. The number of dots to the left of \(x = 3\): \(6+4=10\). The number of dots to the right of \(x = 3\): \(1+2+1=4\).
• The number of dots to the left of \(x = 2\): \(6\). The number of dots to the right of \(x = 2\): \(3+1+2+1=7\).
• Let's calculate the cumulative frequencies properly.
• Order the data (by repeating the \(x\) - values as per the number of dots): \(1,1,1,1,1,1,2,2,2,2,3,3,3,4,5,5,6\)
• The \(9^{th}\) value is \(2\), but if we consider the symmetry (a better measure for center in a dot - plot when we talk about visual center).
• The number of dots:
• Let's count the left - hand side and right - hand side.
• Total dots \(n=17\).
• If we assume the center is a value such that the number of dots on either side is as equal as possible.
• The number of dots to the left of \(x = 3\): \(6 + 4=10\). The number of dots to the right of \(x = 3\): \(1+2+1=4\).
• The number of dots to the left of \(x = 2\): \(6\). The number of dots to the right of \(x = 2\): \(3+1+2+1=7\).
• Let's use the formula for the median of a grouped data (in a sense).
• The data set: \(x_1 = 1,f_1=6\); \(x_2 = 2,f_2 = 4\); \(x_3=3,f_3 = 3\); \(x_4 = 4,f_4=1\); \(x_5 = 5,f_5=2\); \(x_6=6,f_6 = 1\)
• The cumulative frequency \(CF\):
• \(CF_1=6\)
• \(CF_2=6 + 4=10\)
• \(CF_3=10+3 = 13\)
• \(CF_4=13 + 1=14\)
• \(CF_5=14+2=16\)
• \(CF_6=16 + 1=17\)
• The median class (where \(\frac{n}{2}=\frac{17}{2}=8.5\) lies) is the class with \(x = 2\) (since \(CF_1=6\lt8.5\lt CF_2 = 10\)). But if we consider the visual balance (a more intuitive approach for dot - plots)
• Let's count the dots:
• Total dots \(=17\).
• If we consider the middle of the \(17\) dots.
• The \(9^{th}\) dot (when ordered) is \(2\), but if we consider splitting the dot - plot:
• Let's assume we want to split the dot - plot into two parts (left and right) with as close to equal number of dots as possible.
• Number of dots to the left of \(x = 3\): \(6+4=10\). Number of dots to the right of \(x = 3\): \(1+2+1=4\).
• Number of dots to the left of \(x=2\): \(6\). Number of dots to the right of \(x = 2\): \(3 + 1+2+1=7\).
• Let's calculate the mean (another measure of center, \(\bar{x}=\frac{\sum_{i = 1}^{6}x_if_i}{n}=\frac{1\times6+2\times4 + 3\times3+4\times1+5\times2+6\times1}{17}=\frac{6 + 8+9+4+10+6}{17}=\frac{43}{17}\approx2.53\))

Answer:

The dot - plot has \(17\) dots. The dot - plot has \(6\) dots to the left of the center and \(7\) dots to the right of the center. The center of the data set is between \(2\) and \(3\)