Math Problem Statement

How did you calculate the the median show it also do not use interpolation method

Solution

Certainly! Let me explain the process step by step and calculate the median without using interpolation.

Step 1: Understand the Data

We have the following dataCertainly! Let me explain the process step by step and calculate the median without using interpolation.

Step 1: Understand the Data

We have the following

  • Class Intervals: [4,8)[4, 8), [8,12)[8, 12), [12,16)[12, 16), [16,20)[16, 20), [20,24)[20, 24), [24,28)[24, 28), [28,32)[28, 32), [32,36)[32, 36), [36,40)[36, 40), [40,44)[40, 44), [44,48)[44, 48)
  • Frequencies: 8,3,9,5,9,8,4,2,6,3,108, 3, 9, 5, 9, 8, 4, 2, 6, 3, 10

Step 2: Calculate the Cumulative Frequencies

Cumulative frequencies are calculated by adding the frequency of the current class interval to the sum of the frequencies of all previous class intervals.

Here's how the cumulative frequencies look:

  • Cumulative Frequencies: 8,11,20,25,34,42,46,48,54,57,678, 11, 20, 25, 34, 42, 46, 48, 54, 57, 67

Step 3: Find the Median Position

The median is the value that separates the higher half from the lower half of the data. For grouped data, we calculate the median position by:

Median Position=N+12\text{Median Position} = \frac{N + 1}{2} where NN is the total number of observations.

Here, NN is the sum of all frequencies:

N=67N = 67

So, the median position is:

Median Position=67+12=682=34\text{Median Position} = \frac{67 + 1}{2} = \frac{68}{2} = 34

Step 4: Determine the Median Class Interval

The median class interval is the interval where the cumulative frequency reaches or exceeds the median position. From our cumulative frequencies:

  • The cumulative frequency reaches 3434 at the interval [20,24)[20, 24).

Step 5: Determine the Median Without Interpolation

Since the median position of 34 falls within the cumulative frequency of the interval [20,24)[20, 24), we know that the median value lies in this interval. For simplicity, without using interpolation, we can state that the median lies within the range of the class interval:

Median[20,24)\text{Median} \in [20, 24)

Given that this interval is relatively small, a rough estimate of the median would be around the middle of this class interval. Thus, the median value can be taken as:

Median20+242=22\text{Median} \approx \frac{20 + 24}{2} = 22

However, this estimate is less precise than using interpolation, which gives us a more accurate median value.

Would you like more information on any part of this process, or do you have other questions?

Follow-Up Questions:

  1. Why is the median important in understanding the distribution of data?
  2. How would the calculation change if you had a different number of data points?
  3. Can you explain the difference between the median and the mean?
  4. What would happen if the class intervals were of unequal sizes?
  5. How could you find the median if the data were not grouped?

Tip:

When you have grouped data, the median interval provides an important insight into the central tendency, but precise calculation often requires interpolation, especially when the data is not symmetrically distributed.

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Math Problem Analysis

Mathematical Concepts

Statistics
Median
Grouped Data

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12