use the display of data to find the mean, median, mode, and midrange. the mean of the data is . (round to…

Explanation:

Step1: Estimate frequencies

Assume class - midpoints: For the class 10 - 20, mid - point $x_1 = 15$; 20 - 30, mid - point $x_2=25$; 30 - 40, mid - point $x_3 = 35$; 40 - 50, mid - point $x_4=45$. Estimate frequencies: Let $f_1 = 5$, $f_2 = 6$, $f_3 = 7$, $f_4 = 5$. The total number of data points $n=f_1 + f_2+f_3 + f_4=5 + 6+7 + 5=23$.

Step2: Calculate the mean

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{k}f_ix_i}{n}$. So, $\sum_{i = 1}^{4}f_ix_i=f_1x_1 + f_2x_2+f_3x_3 + f_4x_4=5\times15+6\times25 + 7\times35+5\times45=75 + 150+245 + 225=700$. Then $\bar{x}=\frac{700}{23}\approx30.4$.

Step3: Calculate the median

Since $n = 23$ (odd), the median is the $\left(\frac{n + 1}{2}\right)$-th value. $\frac{23+1}{2}=12$-th value. Cumulative frequencies: $CF_1 = 5$, $CF_2=5 + 6 = 11$, $CF_3=5 + 6+7 = 18$. The median class is 30 - 40. Using the median formula $Median = L+\frac{\frac{n}{2}-CF}{f}\times w$, where $L = 30$ (lower limit of median class), $CF = 11$ (cumulative frequency of the previous class), $f = 7$ (frequency of median class), $w = 10$ (class - width). $Median=30+\frac{\frac{23}{2}-11}{7}\times10=30+\frac{0.5}{7}\times10=30+\frac{5}{7}\approx30.7$.

Step4: Determine the mode

The class with the highest frequency is 30 - 40. The mode formula for grouped data is $Mode = L+\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\times w$, where $L = 30$ (lower limit of the modal class), $f_1 = 7$ (frequency of the modal class), $f_0 = 6$ (frequency of the previous class), $f_2 = 5$ (frequency of the next class). $Mode=30+\frac{7 - 6}{2\times7-6 - 5}\times10=30+\frac{1}{3}\times10\approx33.3$.

Step5: Calculate the mid - range

The mid - range is calculated as $\frac{\text{Minimum value}+\text{Maximum value}}{2}$. The minimum value is 10 and the maximum value is 50. So, Mid - range $=\frac{10 + 50}{2}=30$.

Answer:

The mean of the data is $30.4$. The median of the data is $30.7$. A. The mode of the data is $33.3$. The midrange of the data is $30$.