Problem: Find the mean, median, and mode of the sequence \( 19, 12, 9, 5, 8, 5, 25, 8 \).
####
Step 1: Calculate the Mean
The mean (average) of a sequence is calculated using the formula:
\[
\text{Mean} = \frac{\sum x}{N}
\]
where:
- \( \sum x \) is the sum of all the numbers in the sequence.
- \( N \) is the total number of numbers in the sequence.
#####
Given Sequence: \( 19, 12, 9, 5, 8, 5, 25, 8 \)
1.
Sum of the Numbers:
\[
\sum x = 19 + 12 + 9 + 5 + 8 + 5 + 25 + 8 = 81
\]
2.
Total Number of Numbers:
\[
N = 8
\]
3.
Calculate the Mean:
\[
\text{Mean} = \frac{\sum x}{N} = \frac{81}{8} = 10.125
\]
So, the mean is:
\[
\boxed{10.125}
\]
---
####
Step 2: Calculate the Median
The median is the middle value of a sorted sequence. If the sequence has an odd number of elements, the median is the middle element. If the sequence has an even number of elements, the median is the average of the two middle elements.
#####
Given Sequence: \( 19, 12, 9, 5, 8, 5, 25, 8 \)
1.
Sort the Sequence in Ascending Order:
\[
5, 5, 8, 8, 9, 12, 19, 25
\]
2.
Determine the Total Number of Elements:
\[
N = 8 \quad (\text{even number})
\]
3.
Find the Middle Terms:
For an even number of elements, the median is the average of the \( \left(\frac{N}{2}\right)^{\text{th}} \) term and the \( \left(\frac{N}{2} + 1\right)^{\text{th}} \) term.
\[
\frac{N}{2} = \frac{8}{2} = 4 \quad \text{and} \quad \frac{N}{2} + 1 = 4 + 1 = 5
\]
4.
Identify the 4th and 5th Terms:
From the sorted sequence \( 5, 5, 8, 8, 9, 12, 19, 25 \):
- The 4th term is \( 8 \).
- The 5th term is \( 9 \).
5.
Calculate the Median:
\[
\text{Median} = \frac{\text{4th term} + \text{5th term}}{2} = \frac{8 + 9}{2} = \frac{17}{2} = 8.5
\]
So, the median is:
\[
\boxed{8.5}
\]
---
####
Step 3: Calculate the Mode
The mode is the number that appears most frequently in the sequence.
#####
Given Sequence: \( 19, 12, 9, 5, 8, 5, 25, 8 \)
1.
Count the Frequency of Each Number:
\[
\begin{aligned}
&5 \text{ appears } 2 \text{ times}, \\
&8 \text{ appears } 2 \text{ times}, \\
&9 \text{ appears } 1 \text{ time}, \\
&12 \text{ appears } 1 \text{ time}, \\
&19 \text{ appears } 1 \text{ time}, \\
&25 \text{ appears } 1 \text{ time}.
\end{aligned}
\]
2.
Identify the Most Frequent Number(s):
Both \( 5 \) and \( 8 \) appear twice, which is more frequent than any other number.
Since there are two numbers with the highest frequency, the sequence is
bimodal.
So, the modes are:
\[
\boxed{5 \text{ and } 8}
\]
---
Final Answers:
\[
\boxed{10.125, 8.5, 5 \text{ and } 8}
\]
Parent Tip: Review the logic above to help your child master the concept of mean median mode definitions examples.
