Question 1175748: Use the appropriate formula to find the mean,The madian,and the mode of each grouped data.
scores: Frequency:
46-50. 2
41-45. 9
36-40. 10
31-35. 14
26-30. 10
21-25. 5
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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
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