To estimate quartiles from a histogram, you need to identify cumulative frequencies, which will help you determine the positions of the quartiles.

Quartiles divide your data into four equal segments. Specifically, the first quartile (Q1) represents the value below which 25% of the data falls, the second quartile (Q2), or median, is the value below which 50% of the data falls, and the third quartile (Q3) indicates the value below which 75% of the data falls.

To begin, calculate the total number of data points by summing the frequencies of all the bars in the histogram. Next, compute the cumulative frequency for each bar, which represents the running total of frequencies up to that bar. For instance, if the first bar has a frequency of $5$ and the second bar has a frequency of $10$, the cumulative frequency for the second bar is $15$.

To find Q1, locate the position of the 25th percentile in the cumulative frequency. If the total number of data points is $100$, Q1 will be at the $25^{th}$ data point. Check the cumulative frequencies to determine which bar contains this data point. If the $25^{th}$ data point falls within a particular bar, you can estimate Q1 by interpolating within that bar.

Repeat this process to find Q2 (the median) and Q3. For Q2, you will be looking for the $50^{th}$ percentile, while for Q3, you will seek the $75^{th}$ percentile. Again, utilize the cumulative frequencies to identify which bar contains these data points, and perform interpolation within the bar as needed.

By following these steps, you can effectively estimate the quartiles from a histogram, providing valuable insights into the distribution of your data.