Question

descriptive statistics
approximating the mean of a data set given a histogram
the daily online newspaper america at a glance has just released the results of a study of 49 students who completed preparation programs for a nationwide
standardized test. the following histogram, which summarizes the mathematics test score information for the 49 students, appeared in the article
from the histogram, estimate the mean mathematics test score for the students in the study. carry your intermediate computations to at least four decimal
places, and round your answer to at least one decimal place.

Explanation:

Step1: Find the mid - point of each class

Assume the class intervals:

• For the first bar (frequency = 4): If we assume the class is \(550 - 600\), the mid - point \(x_1=\frac{550 + 600}{2}=575\)
• For the second bar (frequency = 11): If the class is \(600 - 650\), the mid - point \(x_2=\frac{600+650}{2}=625\)
• For the third bar (frequency = 14): If the class is \(650 - 700\), the mid - point \(x_3=\frac{650 + 700}{2}=675\)
• For the fourth bar (frequency = 11): If the class is \(700 - 750\), the mid - point \(x_4=\frac{700+750}{2}=725\)
• For the fifth bar (frequency = 9): If the class is \(750 - 800\), the mid - point \(x_5=\frac{750 + 800}{2}=775\)

Step2: Use the formula for the mean of a grouped data \(\bar{x}=\frac{\sum_{i = 1}^{n}f_ix_i}{\sum_{i=1}^{n}f_i}\)

Here, \(f_1 = 4\), \(f_2=11\), \(f_3 = 14\), \(f_4=11\), \(f_5 = 9\) and \(x_1 = 575\), \(x_2=625\), \(x_3=675\), \(x_4=725\), \(x_5=775\)

First, calculate \(\sum_{i = 1}^{n}f_ix_i\):

$$
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And \(\sum_{i=1}^{n}f_i=4 + 11+14+11+9=49\)

Then \(\bar{x}=\frac{33575}{49}\approx685.2\)

Answer:

\(685.2\)