Question

every year, the students at a school are given a musical aptitude test that rates them from 0 (no musical aptitude) to 5 (high musical aptitude). this years results were: aptitude score 0 1 2 3 4 5 frequency 3 6 5 5 2 2 the average (mean) aptitude score: the median aptitude score:

Explanation:

Step1: Calculate the sum of products

First, find the product of each score and its frequency:
For score 0: $0\times3 = 0$
For score 1: $1\times6=6$
For score 2: $2\times5 = 10$
For score 3: $3\times5=15$
For score 4: $4\times2 = 8$
For score 5: $5\times2=10$
The sum of products $\sum_{i = 0}^{5}x_if_i=0 + 6+10 + 15+8+10=49$

Step2: Calculate the total frequency

The total frequency $n=3 + 6+5+5+2+2=23$

Step3: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 0}^{5}x_if_i}{n}=\frac{49}{23}\approx2.13$

Step4: Find the median position

Since $n = 23$ (an odd - numbered data set), the median position is $\frac{n + 1}{2}=\frac{23+1}{2}=12$

Step5: Determine the median

Count the cumulative frequencies:
The cumulative frequency for score 0 is 3.
The cumulative frequency for score 1 is $3 + 6=9$.
The cumulative frequency for score 2 is $9+5 = 14$. Since the 12th value falls within the group of score 2, the median is 2.

Answer:

The average (mean) aptitude score: $\approx2.13$
The median aptitude score: 2