Question

the following scores are written in ascending order: a, b, c, d, e, f, g, h, and i.
a. what measure of central tendency does score e represent?
b. what measure of central tendency is represented by \\(\frac{a + b + c + d + e + f + g + h + i}{9}\\)?
c. which quartile is represented by \\(\frac{b + c}{2}\\)?

Explanation:

Part a

Step1: Recall median definition

The median of a set with \( n \) ordered values (ascending/descending) is the middle value when \( n \) is odd. Here, there are 9 scores (\( a,b,c,d,e,f,g,h,i \)), so \( n = 9 \) (odd). The middle position is \( \frac{9 + 1}{2}=5^{\text{th}} \) term, which is \( e \).

Step1: Recall mean definition

The mean (arithmetic mean) of a set of values \( x_1,x_2,\dots,x_n \) is given by \( \frac{\sum_{i = 1}^{n}x_i}{n} \). Here, the values are \( a,b,c,d,e,f,g,h,i \) (9 values), so \( \frac{a + b + c + d + e + f + g + h + i}{9} \) is the mean.

Step1: Recall quartile definitions

• Quartiles divide data into 4 parts. For \( n = 9 \) data points, the first quartile (\( Q_1 \)) is the median of the lower half (first 4 values: \( a,b,c,d \)). The lower half has 4 values, so median of lower half (for even count) is \( \frac{\text{2nd term}+\text{3rd term}}{2} \). The 2nd term of lower half is \( b \), 3rd term is \( c \), so \( \frac{b + c}{2} \) is \( Q_1 \) (first quartile).

Answer:

Median

Part b