14. Considering the following distribution Calculate: A) median B) All quartiles. C) The 5^ (th ) and 8^ (th ) decile. D) The 50^ (th ) and 80^ (th ) percentile. Values & Frequency 140-150 & 17 150-160 & 29 160-170 & 42 170-180 & 72 180-190 & 84 190-200 & 107 5 Do with a group of 5 members! 200-210 & 49 210-220 & 34 220-230 & 31 230-240 & 16 240-250 & 12

## Step 1: Calculate the Total Frequency<br />### Sum all the frequencies to find the total number of observations. The frequencies are: 17, 29, 42, 72, 84, 107, 49, 34, 31, 16, and 12. Adding these gives a total frequency \( N \).<br /><br />\[<br />N = 17 + 29 + 42 + 72 + 84 + 107 + 49 + 34 + 31 + 16 + 12 = 493<br />\]<br /><br />## Step 2: Calculate the Median<br />### The median is the value that divides the dataset into two equal halves. For grouped data, the median can be found using the formula:<br /><br />\[<br />\text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times w<br />\]<br /><br />Where:<br />- \( L \) is the lower boundary of the median class.<br />- \( CF \) is the cumulative frequency before the median class.<br />- \( f \) is the frequency of the median class.<br />- \( w \) is the class width.<br /><br />### Find the Median Class<br />- \( \frac{N}{2} = \frac{493}{2} = 246.5 \)<br />- Locate the class where the cumulative frequency exceeds 246.5. This occurs in the class 190-200.<br /><br />### Calculate the Median<br />- \( L = 190 \), \( CF = 244 \) (cumulative frequency up to 180-190), \( f = 107 \), \( w = 10 \)<br /><br />\[<br />\text{Median} = 190 + \left( \frac{246.5 - 244}{107} \right) \times 10 = 190 + \left( \frac{2.5}{107} \right) \times 10 \approx 190.23<br />\]<br /><br />## Step 3: Calculate Quartiles<br />### First Quartile (\(Q_1\))<br />- \( Q_1 \) position is at \( \frac{N}{4} = \frac{493}{4} = 123.25 \)<br />- Locate the class where the cumulative frequency exceeds 123.25. This occurs in the class 170-180.<br /><br />\[<br />Q_1 = 170 + \left( \frac{123.25 - 118}{72} \right) \times 10 = 170 + \left( \frac{5.25}{72} \right) \times 10 \approx 170.73<br />\]<br /><br />### Third Quartile (\(Q_3\))<br />- \( Q_3 \) position is at \( \frac{3N}{4} = \frac{3 \times 493}{4} = 369.75 \)<br />- Locate the class where the cumulative frequency exceeds 369.75. This occurs in the class 200-210.<br /><br />\[<br />Q_3 = 200 + \left( \frac{369.75 - 351}{49} \right) \times 10 = 200 + \left( \frac{18.75}{49} \right) \times 10 \approx 203.83<br />\]<br /><br />## Step 4: Calculate Deciles<br />### Fifth Decile (\(D_5\)) is the same as the median, so \( D_5 = 190.23 \).<br /><br />### Eighth Decile (\(D_8\))<br />- \( D_8 \) position is at \( \frac{8N}{10} = \frac{8 \times 493}{10} = 394.4 \)<br />- Locate the class where the cumulative frequency exceeds 394.4. This occurs in the class 200-210.<br /><br />\[<br />D_8 = 200 + \left( \frac{394.4 - 351}{49} \right) \times 10 = 200 + \left( \frac{43.4}{49} \right) \times 10 \approx 208.86<br />\]<br /><br />## Step 5: Calculate Percentiles<br />### Fiftieth Percentile (\(P_{50}\)) is the same as the median, so \( P_{50} = 190.23 \).<br /><br />### Eightieth Percentile (\(P_{80}\))<br />- \( P_{80} \) position is at \( \frac{80N}{100} = \frac{80 \times 493}{100} = 394.4 \)<br />- This is the same calculation as for \( D_8 \), so \( P_{80} = 208.86 \).