Question

the frequency tables show the distances traveled (in miles) to work everyday by employees from two different companies. display the data in the histograms. compare the distributions using their shapes and appropriate measures of center and variation.
company a
distance: 0-8, 9-17, 18-26, 27-35, 36-44
frequency: 9, 12, 18, 12, 9
company b
distance: 0-8, 9-17, 18-26, 27-35, 36-44
frequency: 4, 7, 9, 13, 11

Explanation:

Step1: Analyze Shape of Company A

The frequencies for Company A are 9, 12, 18, 12, 9. The frequency increases to the middle (18 at 18 - 26) and then decreases symmetrically. So, Company A's distribution is symmetric (bell - shaped).

Step2: Analyze Shape of Company B

The frequencies for Company B are 4, 7, 9, 13, 11. The frequency generally increases as the distance interval increases (from 0 - 8 to 27 - 35, then slightly decreases at 36 - 44). So, Company B's distribution is skewed left (or has a longer tail on the left) or we can say it is approximately skewed or has a trend of increasing frequency with distance (more employees travel longer distances).

Step3: Measure of Center (Median) for Company A

First, find the total number of employees in Company A: \(n_A=9 + 12+18 + 12+9=60\). The median is the average of the 30th and 31st values.

• Cumulative frequency for 0 - 8: 9
• Cumulative frequency for 9 - 17: \(9 + 12 = 21\)
• Cumulative frequency for 18 - 26: \(21+18 = 39\)

Since 30 and 31 fall in the 18 - 26 interval, the median class is 18 - 26. We can also note that due to symmetry, the center (median) is around the middle interval (18 - 26).

Step4: Measure of Center (Median) for Company B

Total number of employees in Company B: \(n_B=4 + 7+9 + 13+11 = 44\). The median is the 22nd value.

• Cumulative frequency for 0 - 8: 4
• Cumulative frequency for 9 - 17: \(4 + 7=11\)
• Cumulative frequency for 18 - 26: \(11 + 9 = 20\)
• Cumulative frequency for 27 - 35: \(20+13 = 33\)

The 22nd value falls in the 27 - 35 interval. So, the median class for Company B is 27 - 35, which is to the right of the median class of Company A. This means that on average, employees in Company B travel longer distances to work than those in Company A.

Step5: Measure of Variation (Inter - quartile Range or Range)

For Company A, the data is symmetric. The range is \(44 - 0=44\) (since the maximum distance interval is 36 - 44 (upper limit can be considered as 44) and minimum is 0 - 8 (lower limit 0)). The inter - quartile range: First quartile (\(Q_1\)) is the 15th value (since \(n_A = 60\), \(Q_1\) is at \(\frac{60}{4}=15\)th value) and third quartile (\(Q_3\)) is at \(45\)th value.

• Cumulative frequency for 0 - 8: 9, 9 - 17: 21. So \(Q_1\) is in 9 - 17 (since 15 is between 9 and 21).
• Cumulative frequency for 18 - 26: 39, 27 - 35: 51, 36 - 44: 60. \(Q_3\) is in 27 - 35 (45 is between 39 and 51). The IQR is the difference between \(Q_3\) and \(Q_1\) intervals. Since the distribution is symmetric, the variation is relatively balanced.

For Company B, the range is also \(44 - 0 = 44\). The first quartile (\(Q_1\)) is at \(\frac{44}{4}=11\)th value. Cumulative frequency for 0 - 8: 4, 9 - 17: 11. So \(Q_1\) is at the end of 9 - 17. The third quartile (\(Q_3\)) is at \(33\)rd value. Cumulative frequency for 27 - 35: 33. So \(Q_3\) is at the end of 27 - 35. The IQR for Company B: The difference between the upper quartile and lower quartile intervals. Since the distribution is skewed, the variation is such that more data is on the higher end, but the range is the same as Company A. However, the median of Company B is higher than that of Company A, indicating that Company B's employees travel longer distances on average. Also, Company A's distribution is symmetric, while Company B's is skewed (towards longer distances).

Answer:

• Shape: Company A has a symmetric (bell - shaped) distribution. Company B has a distribution that is skewed left (or has a trend of increasing frequency with distance, more employees travel longer distances).
• Measure of Center (Median): The median distance for Company A falls in the 18 - 26 mile interval. The median distance for Company B falls in the 27 - 35 mile interval. This means employees at Company B travel longer distances to work on average than those at Company A.
• Measure of Variation: Both companies have the same range (from 0 to 44 miles). For Company A (symmetric), the data is more evenly distributed around the center. For Company B (skewed), the data is more concentrated on the higher - distance end, but the overall range is the same.