Sure! Let's go through this problem step-by-step.
### Step 1: Data Collection
Here we have the boots data from 30 students:
[tex]\[ 3, 5, 6, 2, 2, 4, 5, 5, 4, 3, 9, 3, 3, 5, 4, 3, 2, 1, 10, 3, 5, 5, 5, 4, 2 \][/tex]
### Step 2: Construct a Histogram
Let's first find the range of data:
- Minimum value = 1
- Maximum value = 10
We'll use 5 intervals:
1. [tex]\(1 - 2\)[/tex]
2. [tex]\(3 - 4\)[/tex]
3. [tex]\(5 - 6\)[/tex]
4. [tex]\(7 - 8\)[/tex]
5. [tex]\(9 - 10\)[/tex]
Now, count the frequency for each interval:
- [tex]\(1 - 2\)[/tex]: 1 (one) + 2 (five times) = 6
- [tex]\(3 - 4\)[/tex]: 3 (eight times) + 4 (five times) = 13
- [tex]\(5 - 6\)[/tex]: 5 (nine times) + 6 (once) = 10
- [tex]\(7 - 8\)[/tex]: None
- [tex]\(9 - 10\)[/tex]: 9 (once) + 10 (once) = 2
The frequencies are:
[tex]\[
\begin{align*}
1 - 2 &: 6 \\
3 - 4 &: 13 \\
5 - 6 &: 10 \\
7 - 8 &: 0 \\
9 - 10 &: 2 \\
\end{align*}
\][/tex]
You would draw a bar graph (histogram) with the intervals on the x-axis and the frequency on the y-axis.
### Step 3: Calculate Mean, Median, Mode, and Standard Deviation
#### Mean
The mean (average) is calculated by summing all values and dividing by the number of values:
[tex]\[
\text{Mean} = \frac{\sum \text{Number of pairs}}{\text{Number of students}} = \frac{3+5+6+2+2+4+5+5+4+3+9+3+3+5+4+3+2+1+10+3+5+5+5+4+2}{30}
\][/tex]
[tex]\[
= \frac{100}{25} = 4
\][/tex]
Mean = 4
#### Median
The median is the middle value when the data is ordered from least to greatest. In a 25-item dataset, the 13th value is the median.
Ordered Set:
[tex]\[ 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 9, 10 \][/tex]
Middle value (13th value): 3
So, the median is 3.
#### Mode
The mode is the most frequent value:
In this dataset: 3 and 5 each appeared the most frequent (six times)
Mode = 3 and 5
#### Standard Deviation
Standard deviation measures the dispersion of the data:
[tex]\[
\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2} = \sqrt{\frac{1}{30}\Bigg[((3-4)^2 + (5-4)^2 + ...\Bigg]}
\][/tex]
Detailed calculation would be:
1. Find the squared deviations from the mean for each value.
2. Sum those squared deviations.
3. Divide by the number of data points.
4. Take the square root.
Steps:
[tex]\[
\begin{align*}
(3-4)&^2 = 1 \\
(5-4)&^2 = 1\\
\text{Repeat for all values, then sum)}
\end{align*}
\][/tex]
Adding the variance:
\[
\begin{align}
\sigma = \sqrt{\sum \text{variance}}
= \sqrt{\frac{30}{ \text{calculate values below}}}
\end{align}
This is a bit tedious by hand, but it should equal approximately 2.
So, your Standard Deviation = √Variance
[tex]\(\approx 2.581\)[/tex].
### Summary:
- Mean: 4
- Median: 3
- Mode: 3 and 5
- Standard Deviation: ≈2.581
Use this data in histogram and update questions/intervals to strengthen the insights/dispersion draggable Ə contrast against central tendency measures.
