Source: The World Almanac and Book of Facts. 9. Weights of the NBA's Top \( \mathbf{5 0} \) Players Listed are the weights of the NBA's top 50 players. Construct a grouped frequency distribution and a cumulative frequency distribution with 8 classes. Analyze the results in terms of peaks, extreme values, etc. \[ \begin{array}{llllllllll} 240 & 210 & 220 & 260 & 250 & 195 & 230 & 270 & 325 & 2251 \\ \hline 165 & 295 & 205 & 230 & 250 & 210 & 220 & 210 & 230 & 202 \\ 250 & 265 & 230 & 210 & 240 & 245 & 225 & 180 & 175 & 215 \\ 215 & 235 & 245 & 250 & 215 & 210 & 195 & 240 & 240 & 225 \\ 260 & 210 & 190 & 260 & 230 & 190 & 210 & 230 & 185 & 260 \\ \hline \end{array} \]

Certainly! Let's analyze the weights of the NBA's top 50 players by constructing both a grouped frequency distribution and a cumulative frequency distribution with 8 classes. We'll also discuss the observed patterns, peaks, and extreme values. --- ## **1. Data Preparation** ### **Original Data:** ``` Row 1: 240, 210, 220, 260, 250, 195, 230, 270, 325, 2251 Row 2: 165, 295, 205, 230, 250, 210, 220, 210, 230, 202 Row 3: 250, 265, 230, 210, 240, 245, 225, 180, 175, 215 Row 4: 215, 235, 245, 250, 215, 210, 195, 240, 240, 225 Row 5: 260, 210, 190, 260, 230, 190, 210, 230, 185, 260 ``` ### **Data Correction:** - **Note:** The value **2251** in Row 1 appears to be a typo. It's assumed to be **251** for a realistic weight range. ### **Corrected Data:** ``` 240, 210, 220, 260, 250, 195, 230, 270, 251, 165, 295, 205, 230, 250, 210, 220, 210, 230, 202, 250, 265, 230, 210, 240, 245, 225, 180, 175, 215, 215, 235, 245, 250, 215, 210, 195, 240, 240, 225, 260, 210, 190, 260, 230, 190, 210, 230, 185, 260 ``` ### **Sorted Data:** ``` 165, 175, 180, 185, 190, 190, 195, 195, 202, 205, 210, 210, 210, 210, 210, 210, 215, 215, 215, 215, 220, 220, 225, 225, 225, 230, 230, 230, 230, 230, 230, 235, 240, 240, 240, 240, 240, 245, 245, 250, 250, 250, 250, 251, 260, 260, 260, 260, 265, 270, 295 ``` --- ## **2. Grouped Frequency Distribution** ### **Determining Classes:** - **Range:** \( 295 - 165 = 130 \) lbs - **Number of Classes:** 8 - **Class Width:** \( \frac{130}{8} = 16.25 \) lbs → Rounded to **20** lbs for simplicity. ### **Class Intervals:** | Class Interval (lbs) | Frequency | |----------------------|-----------| | 160 – 179 | 2 | | 180 – 199 | 6 | | 200 – 219 | 13 | | 220 – 239 | 10 | | 240 – 259 | 12 | | 260 – 279 | 6 | | 280 – 299 | 1 | | 300 – 319 | 0 | **_Frequency Distribution Table:_** | **Weight Range (lbs)** | **Frequency** | |------------------------|---------------| | 160 – 179 | 2 | | 180 – 199 | 6 | | 200 – 219 | 13 | | 220 – 239 | 10 | | 240 – 259 | 12 | | 260 – 279 | 6 | | 280 – 299 | 1 | | 300 – 319 | 0 | | **Total** | **50** | --- ## **3. Cumulative Frequency Distribution** | Class Interval (lbs) | Frequency | Cumulative Frequency | |----------------------|-----------|----------------------| | 160 – 179 | 2 | 2 | | 180 – 199 | 6 | 8 | | 200 – 219 | 13 | 21 | | 220 – 239 | 10 | 31 | | 240 – 259 | 12 | 43 | | 260 – 279 | 6 | 49 | | 280 – 299 | 1 | 50 | | 300 – 319 | 0 | 50 | --- ## **4. Analysis of Results** ### **Peaks:** - **200 – 219 lbs:** Highest frequency with **13** players. This suggests a significant concentration of players in this weight range. - **240 – 259 lbs:** Second peak with **12** players, indicating another common weight range among top players. ### **Moderate Frequencies:** - **220 – 239 lbs:** **10** players fall into this category, bridging the two main peaks. ### **Lower Frequencies:** - **180 – 199 lbs:** **6** players. Fewer players are in the lighter weight brackets. - **260 – 279 lbs:** **6** players. Similar to the lighter range, fewer players are in the higher weight brackets. ### **Extreme Values:** - **Minimum Weight:** **165 lbs** – Represents the lightest player in the top 50. - **Maximum Weight:** **295 lbs** – Represents the heaviest player in the top 50. ### **Outliers:** - **295 lbs:** A notable outlier on the higher end, indicating at least one exceptionally heavy player compared to peers. ### **No Data:** - **300 – 319 lbs:** **0** players, confirming that this weight range is beyond the scope of the top 50. ### **Overall Insights:** - The distribution is **right-skewed**, with a concentration of players in the lower to mid-weight ranges and fewer players as the weight increases. - The two prominent peaks suggest two common body types or playing styles among top NBA players. - The presence of outliers indicates that while most players fall within certain weight ranges, there are exceptions that stand out. --- ## **5. Visual Representation (Optional)** For a clearer understanding, constructing histograms or bar charts based on the frequency distribution can visually highlight the peaks and distribution spread. --- ## **Conclusion** The grouped frequency distribution for the NBA's top 50 players' weights reveals two main clusters around **200-219 lbs** and **240-259 lbs**, with fewer players in both lighter and heavier categories. The extremes of **165 lbs** and **295 lbs** showcase the range of player physiques, although the majority fall within the identified peaks. This distribution can aid in understanding the common body metrics associated with elite basketball performance.