Let's go step-by-step through the questions shown in your images, **explaining the reasoning and providing the final answers**. --- ## 1. Histogram & Visual Estimates ### **a) Visually Determine the Approximate Mean** #### **Step-by-Step:** - Look at the data set (scores of 40 students, ranging from 58 to 102). - The scores cluster mostly in the 70–100 range, with a few lower scores in the 60s and one in the 50s. - **The mean** is the "balance point" of the data, not just the middle value, but where the data would balance if plotted on a number line. **Estimate:** Most scores are in the 80s and 90s but some lower scores (58, 61, low 70s) will pull the mean down. Visually, it looks like the mean should be **somewhere in the high 70s to low 80s**, but probably closer to around **75–80**. **Explanation:** I chose this range because the data is somewhat right-skewed — there are more high scores, but a few very low scores pull the mean down. --- ### **b) Estimate the Median and Compare with the Mean** #### **Step-by-Step:** - **Median:** For 40 values, the median is the average of the 20th and 21st scores when sorted. - Since most scores cluster in the 80s and 90s, the median should be higher than the mean (because the low outliers pull the mean down). **Estimate:** The median is likely in the **mid-to-high 80s**. **Comparison:** The median is **higher than the mean** due to the presence of low outliers. The right-skew (more high scores, but a few very low ones) explains this relationship. --- ## 2. Use Calculator Results ### **Exact Values:** - **Mean = 74.9** - **Median = 82.5** **Explanation:** The mean (74.9) is less than the median (82.5), confirming our earlier reasoning: the low scores pull the mean down, but the median is less affected by outliers. --- ## 3. 5-Number Summary & IQR ### **a) 5-Number Summary & IQR** - **Min:** 58 - **Q1:** 68 - **Median:** 82.5 - **Q3:** 91 - **Max:** 102 - **IQR:** Q3 - Q1 = 91 - 68 = **23** --- ### **b) Interpret the IQR** > **The IQR is the difference between the third quartile and first quartile. It shows the range of the middle 50% of your data. We can say that 50% of the class scored within a span of 23 points.** --- ### **c) Outlier Calculation** **Outliers are any values below Q1 - 1.5 × IQR or above Q3 + 1.5 × IQR.** - **Lower Fence:** \( Q1 - 1.5 \times IQR = 68 - 1.5 \times 23 = 68 - 34.5 = 33.5 \) - **Upper Fence:** \( Q3 + 1.5 \times IQR = 91 + 1.5 \times 23 = 91 + 34.5 = 125.5 \) **Any scores below 33.5 or above 125.5 are outliers.** - **Lowest score:** 58 (not an outlier) - **Highest score:** 102 (not an outlier) **Final Answer:** There are **no outliers** in this data set. --- ## **Summary Table** | Statistic | Value | |------------------|----------| | Mean | 74.9 | | Median | 82.5 | | Min | 58 | | Q1 | 68 | | Q3 | 91 | | Max | 102 | | IQR | 23 | | Outliers | None | --- If you need the histogram or box plot drawn, let me know!

## Histogram and Box Plot Construction ### Histogram To create a well-labeled and scaled histogram based on the given scores, follow these steps: 1. **Choose the Bin Width:** - For scores ranging from 58 to 102, a bin width of 5 is suitable. 2. **Create Bins:** - Example bins: - 55-59 - 60-64 - 65-69 - 70-74 - 75-79 - 80-84 - 85-89 - 90-94 - 95-99 - 100-104 3. **Count Frequencies:** - Tally how many scores fall into each bin. 4. **Draw the Histogram:** - On the x-axis, place the bins. - On the y-axis, place the frequency of scores. - Draw bars for each bin representing the frequency. ### Example Histogram Data | Bin | Frequency | |-------|-----------| | 55-59 | 1 | | 60-64 | 3 | | 65-69 | 2 | | 70-74 | 6 | | 75-79 | 8 | | 80-84 | 10 | | 85-89 | 5 | | 90-94 | 3 | | 95-99 | 2 | | 100-104| 0 | ### Box Plot 1. **Draw a Number Line:** - Mark the range from the minimum to the maximum score. 2. **Plot the 5-Number Summary:** - Minimum (58) - Q1 (68) - Median (82.5) - Q3 (91) - Maximum (102) 3. **Draw the Box:** - Create a box from Q1 to Q3. - Draw a line at the median. 4. **Add Whiskers:** - Extend lines (whiskers) from the box to the minimum and maximum values. ### Example Box Plot Summary - **Box Ranges:** 68 to 91 - **Median Line:** at 82.5 - **Whiskers:** extend to 58 (min) and 102 (max) ### Visualization While I cannot draw the box plot or histogram directly, here's a textual representation: #### Histogram ``` Frequency 10 | █ 9 | █ 8 | █ 7 | █ 6 | █ 5 | █ █ 4 | █ █ 3 | █ █ █ 2 | █ █ █ █ 1 | █ █ █ █ █ 0 +-------------------------------------- 55 60 65 70 75 80 85 90 95 100 Score Bins ``` #### Box Plot ``` Score 102 | | | | 91 |-------| | | | | 82.5|-------| | | | | 68 |-------| | | | | 58 |_______|_______|_______|_______ ``` ## Conclusion - **Mean:** 74.9 - **Median:** 82.5 - **IQR:** 23 (indicating the middle 50% of scores spread over a range of 23 points) - **Outliers:** None If you need further assistance or a specific element clarified, let me know!