Question

question 1-6
5 points
given the data set below, find the mean, median, mode, range, and iqr
1,1,2,3,4,5,7,8,9,9,9,14,25
mean:
median:
mode:
range:
iqr:
use the box below or your capture sheet for your work, explanation, and/or answer for each part. if you are using the capture sheet, please enter in the -see capture sheet.

Explanation:

Step1: Calculate the Mean

Sum the data points: \(1 + 1 + 2 + 3 + 4 + 5 + 7 + 8 + 9 + 9 + 9 + 14 + 25\)
First, add step by step:
\(1+1 = 2\); \(2+2 = 4\); \(4+3 = 7\); \(7+4 = 11\); \(11+5 = 16\); \(16+7 = 23\); \(23+8 = 31\); \(31+9 = 40\); \(40+9 = 49\); \(49+9 = 58\); \(58+14 = 72\); \(72+25 = 97\)
Number of data points \(n = 13\)
Mean \(=\frac{\text{Sum}}{n}=\frac{97}{13}\approx7.46\) (rounded to two decimal places)

Step2: Find the Median

Arrange data in order (already ordered: \(1,1,2,3,4,5,7,8,9,9,9,14,25\))
\(n = 13\) (odd), so median is the \(\frac{13 + 1}{2}=7^{\text{th}}\) term.
Count to the 7th term: 1 (1st), 1 (2nd), 2 (3rd), 3 (4th), 4 (5th), 5 (6th), 7 (7th). So median is 7.

Step3: Determine the Mode

Mode is the most frequent value. 9 appears 3 times, others less. So mode is 9.

Step4: Calculate the Range

Range = Maximum - Minimum = \(25 - 1 = 24\)

Step5: Calculate the IQR

First, find Q1 (25th percentile) and Q3 (75th percentile).
For \(n = 13\), the median is the 7th term (7). The lower half is data before the median: \(1,1,2,3,4,5\) (6 terms), upper half: \(8,9,9,9,14,25\) (6 terms).
Q1 is the median of the lower half: \(\frac{3 + 4}{2}=3.5\) (since 6 terms, average of 3rd and 4th: 2 (3rd? Wait, no: lower half is \(1,1,2,3,4,5\) – positions 1 - 6. Median of lower half: term at \(\frac{6}{2}=3^{\text{rd}}\) and \(\frac{6}{2}+1 = 4^{\text{th}}\). 3rd term is 2, 4th term is 3. So Q1 \(=\frac{2 + 3}{2}=2.5\)? Wait, no, earlier mistake: original data set count: let's recount the data points:
Data set: 1,1,2,3,4,5,7,8,9,9,9,14,25. Let's list indices 1 - 13:
1:1, 2:1, 3:2, 4:3, 5:4, 6:5, 7:7, 8:8, 9:9, 10:9, 11:9, 12:14, 13:25.
So lower half is indices 1 - 6: [1,1,2,3,4,5] (6 elements). Median of lower half (Q1) is average of 3rd and 4th elements: 3rd is 2, 4th is 3. So Q1 \(=\frac{2 + 3}{2}=2.5\)
Upper half is indices 8 - 13: [8,9,9,9,14,25] (6 elements). Median of upper half (Q3) is average of 3rd and 4th elements: 3rd is 9, 4th is 9. So Q3 \(=\frac{9 + 9}{2}=9\)
IQR = Q3 - Q1 = \(9 - 2.5 = 6.5\)

Answer:

• Mean: \(\approx 7.46\)
• Median: \(7\)
• Mode: \(9\)
• Range: \(24\)
• IQR: \(6.5\)