Question

goal: find the mean, median, and mode of data given in a frequency chart.
here is a frequency distribution table (fdt) for a small data set:
find the following measures of central tendency.
mean (\\( \overline { x } \\) ) = (please show your answer to one decimal place.)
median = (please enter an exact answer.)
mode = (please enter an exact answer.)
question help: message instructor

Explanation:

Step1: Calculate the mean

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}f_{i}}{\sum_{i = 1}^{n}f_{i}}\)
First, calculate \(\sum_{i = 1}^{n}x_{i}f_{i}\):
For \(x = 22\) and \(f=6\), \(x\times f=22\times6 = 132\)
For \(x = 23\) and \(f = 2\), \(x\times f=23\times2=46\)
For \(x = 24\) and \(f = 7\), \(x\times f=24\times7 = 168\)
For \(x = 25\) and \(f = 4\), \(x\times f=25\times4=100\)
For \(x = 26\) and \(f = 3\), \(x\times f=26\times3 = 78\)
\(\sum_{i = 1}^{n}x_{i}f_{i}=132 + 46+168+100+78=524\)
\(\sum_{i = 1}^{n}f_{i}=6 + 2+7+4+3=22\)
\(\bar{x}=\frac{524}{22}\approx23.8\)

Step2: Calculate the median

The cumulative frequencies:
For \(x = 22\), cumulative frequency \(CF_1=6\)
For \(x = 23\), cumulative frequency \(CF_2=6 + 2=8\)
For \(x = 24\), cumulative frequency \(CF_3=8+7 = 15\)
Since \(n = 22\), \(\frac{n}{2}=11\)
The median class is the class where the cumulative frequency is just greater than or equal to \(\frac{n}{2}\). Here, when \(x = 24\), \(CF = 15\geq11\)
The median is \(24\)

Step3: Calculate the mode

The mode is the data - value with the highest frequency.
The frequencies are \(6\) (for \(x = 22\)), \(2\) (for \(x = 23\)), \(7\) (for \(x = 24\)), \(4\) (for \(x = 25\)), \(3\) (for \(x = 26\))
Since the highest frequency is \(7\) corresponding to \(x = 24\), the mode is \(24\)

Answer:

mean \((\bar{x}) = 23.8\)
median \(=24\)
mode \(=24\)