Question 1175748
Absolutely! Let's break down how to find the mean, median, and mode for this grouped data.

**1. Finding the Mean**

* **Midpoints:** First, find the midpoint of each class interval.
    * 46-50: (46 + 50) / 2 = 48
    * 41-45: (41 + 45) / 2 = 43
    * 36-40: (36 + 40) / 2 = 38
    * 31-35: (31 + 35) / 2 = 33
    * 26-30: (26 + 30) / 2 = 28
    * 21-25: (21 + 25) / 2 = 23

* **Multiply Midpoints by Frequencies:** Multiply each midpoint by its corresponding frequency.
    * 48 * 2 = 96
    * 43 * 9 = 387
    * 38 * 10 = 380
    * 33 * 12 = 396
    * 28 * 7 = 196
    * 23 * 5 = 115

* **Sum of (Midpoint * Frequency):** Add up the results from the previous step.
    * 96 + 387 + 380 + 396 + 196 + 115 = 1570

* **Sum of Frequencies:** Add up all the frequencies.
    * 2 + 9 + 10 + 12 + 7 + 5 = 45

* **Calculate the Mean:** Divide the sum of (midpoint * frequency) by the sum of frequencies.
    * Mean = 1570 / 45 ≈ 34.89

**2. Finding the Median**

* **Cumulative Frequency:** Calculate the cumulative frequency.
    * 2
    * 2 + 9 = 11
    * 11 + 10 = 21
    * 21 + 12 = 33
    * 33 + 7 = 40
    * 40 + 5 = 45

* **Median Position:** Find the median position.
    * Median position = (Total frequency + 1) / 2 = (45 + 1) / 2 = 23

* **Median Class:** Identify the class interval containing the median position. The 23rd value falls within the 36-40 class (cumulative frequency 21 to 33)

* **Median Formula (for grouped data):**
    * Median = L + [(N/2 - CF) / f] * w
    * Where:
        * L = Lower boundary of the median class (35.5)
        * N = Total frequency (45)
        * CF = Cumulative frequency of the class before the median class (21)
        * f = Frequency of the median class (12)
        * w = Class width (5)
    * Median = 35.5 + [(45/2 - 21) / 12] * 5
    * Median = 35.5 + [(22.5 - 21) / 12] * 5
    * Median = 35.5 + (1.5 / 12) * 5
    * Median = 35.5 + (0.125) * 5
    * Median = 35.5 + 0.625
    * Median = 36.125

**3. Finding the Mode**

* **Modal Class:** Identify the class interval with the highest frequency. In this case, the 31-35 class has the highest frequency (12).

* **Mode Formula (for grouped data):**
    * Mode = L + [(d1) / (d1 + d2)] * w
    * Where:
        * L = Lower boundary of the modal class (30.5)
        * d1 = Frequency of the modal class - Frequency of the class before the modal class (12 - 10 = 2)
        * d2 = Frequency of the modal class - Frequency of the class after the modal class (12 - 7 = 5)
        * w = Class width (5)
    * Mode = 30.5 + [2 / (2 + 5)] * 5
    * Mode = 30.5 + (2 / 7) * 5
    * Mode = 30.5 + (0.2857) * 5
    * Mode = 30.5 + 1.4285
    * Mode = 31.9285

**Summary**

* Mean: ≈ 34.89
* Median: ≈ 36.125
* Mode: ≈ 31.9285