To calculate the midpoint ($x$) for each interval, you take the average of the lower and upper boundaries of that interval. Here’s the step-by-step process:

1. Identify the interval boundaries: For example, for the interval $0 \leq T < 20$, the boundaries are 0 and 20.

2. Calculate the midpoint: Add the lower and upper boundaries, then divide by 2 to get the average. This gives the central value for that interval, which is the midpoint.

   $x = \frac{\text{Lower Boundary} + \text{Upper Boundary}}{2}$

3. Apply to each interval: For each interval in the table, calculate the midpoint in this way. Here are the calculations based on the image:

   • Interval $0 \leq T < 20$: $x = \frac{0 + 20}{2} = 10$
   • Interval $20 \leq T < 40$: $x = \frac{20 + 40}{2} = 30$
   • Interval $40 \leq T < 60$: $x = \frac{40 + 60}{2} = 50$
   • Interval $60 \leq T < 80$: $x = \frac{60 + 80}{2} = 70$
   • Interval $80 \leq T < 100$: $x = \frac{80 + 100}{2} = 90$
   • Interval $100 \leq T < 120$: $x = \frac{100 + 120}{2} = 110$

These midpoint values are listed in the table under the $x$ column for each interval. Midpoints are used to represent the average value of each interval for further calculations, like finding the estimated mean.

Would you like further details on this or have any other questions?

Here are five related questions to deepen your understanding:

1. How is the estimated mean calculated using midpoints and frequencies?
2. Why is the midpoint important when estimating means for grouped data?
3. Can midpoints be used to approximate the median or mode?
4. How would the calculation change if the intervals were unequal?
5. What are other statistical measures you can calculate using midpoints?

Tip: Midpoints are essential when working with grouped data as they represent the typical value of each interval, simplifying calculations for means and other statistics.