Let's analyze the histograms provided in the image to determine the median and quartiles.

Part a:

The data has the following midpoints (with their frequencies):

• 33: frequency 4
• 38: frequency 10
• 43: frequency 12
• 48: frequency 16
• 53: frequency 14
• 58: frequency 10
• 63: frequency 4

Total number of data points:

Summing the frequencies: $4 + 10 + 12 + 16 + 14 + 10 + 4 = 70$

Median:

The median is the middle value in an ordered dataset. For 70 data points, the median lies between the 35th and 36th values.

• The cumulative frequencies:
  • First interval (33): 4
  • Second interval (38): 4 + 10 = 14
  • Third interval (43): 14 + 12 = 26
  • Fourth interval (48): 26 + 16 = 42 (this interval contains the median, as both the 35th and 36th values lie here).

Since the median falls in the interval 48, the exact median can be approximated using interpolation if needed.

Quartiles:

• Q1 (25th percentile): It lies at the 17.5th value (25% of 70 = 17.5).

  • From the cumulative frequencies, it falls in the third interval (43) since the 17th and 18th values lie in this range.
  • So, Q1 ≈ 43.

• Q3 (75th percentile): It lies at the 52.5th value (75% of 70 = 52.5).

  • From the cumulative frequencies, the 52nd and 53rd values lie in the fifth interval (53).
  • So, Q3 ≈ 53.

Part b:

The data has the following midpoints (with their frequencies):

• 12: frequency 2
• 17: frequency 7
• 22: frequency 12
• 27: frequency 16
• 32: frequency 8

Total number of data points:

Summing the frequencies: $2 + 7 + 12 + 16 + 8 = 45$

Median:

For 45 data points, the median lies at the 23rd value.

• The cumulative frequencies:
  • First interval (12): 2
  • Second interval (17): 2 + 7 = 9
  • Third interval (22): 9 + 12 = 21
  • Fourth interval (27): 21 + 16 = 37 (this interval contains the median, as the 23rd value lies here).

So, the median is approximately 27.

Quartiles:

• Q1 (25th percentile): It lies at the 11.25th value (25% of 45 = 11.25).

  • From the cumulative frequencies, the 11th and 12th values lie in the third interval (22).
  • So, Q1 ≈ 22.

• Q3 (75th percentile): It lies at the 33.75th value (75% of 45 = 33.75).

  • From the cumulative frequencies, the 33rd and 34th values lie in the fourth interval (27).
  • So, Q3 ≈ 27.

Would you like more details on the interpolation process or need any further clarifications?

Here are some related questions to explore:

1. How do you calculate exact percentiles from grouped data?
2. What is the difference between the arithmetic mean and the median?
3. How does the shape of a histogram affect the interpretation of the median?
4. Can you always approximate quartiles from a histogram?
5. What are the limitations of using histograms for calculating measures of central tendency?

Tip: The more symmetrical the data distribution is, the closer the mean and median tend to be.