The table shows the height of teak trees harvested by a farmer.
| Height(m) | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|
| Number of trees | 4 | 6 | 4 | 5 | 6 | 2 |
Answer (1)
Calculate the total number of trees: 4 + 6 + 4 + 5 + 6 + 2 = 27 .
Calculate the sum of (height * number of trees): ( 3 × 4 ) + ( 4 × 6 ) + ( 5 × 4 ) + ( 6 × 5 ) + ( 7 × 6 ) + ( 8 × 2 ) = 144 .
Calculate the average height: 27 144 ≈ 5.33 meters.
Calculate the standard deviation: 2.44 ≈ 1.56 meters. The average height is approximately 5.33 meters, and the standard deviation is approximately 1.56 meters.
Explanation
Analyzing the Data First, let's analyze the data provided in the table. We have the heights of teak trees and the number of trees for each height. The heights are 3, 4, 5, 6, 7, and 8 meters, and the corresponding number of trees are 4, 6, 4, 5, 6, and 2.
Objective To better understand the distribution of tree heights, we can calculate some descriptive statistics such as the average height and the standard deviation.
Total Number of Trees The total number of trees is calculated by summing the number of trees for each height: 4 + 6 + 4 + 5 + 6 + 2 = 27 .
Calculating the Weighted Sum of Heights Next, we calculate the sum of (height * number of trees): ( 3 × 4 ) + ( 4 × 6 ) + ( 5 × 4 ) + ( 6 × 5 ) + ( 7 × 6 ) + ( 8 × 2 ) = 12 + 24 + 20 + 30 + 42 + 16 = 144 .
Calculating the Average Height Now, we can calculate the average height by dividing the sum of (height * number of trees) by the total number of trees: 27 144 = 5.333... ≈ 5.33 meters.
Calculating the Variance To calculate the variance, we use the formula: t o t a l _ t rees ∑ ( h e i g h t − a v er a g e _ h e i g h t ) 2 × n u mb er _ o f _ t rees . So, the variance is: 27 ( 3 − 5.33 ) 2 × 4 + ( 4 − 5.33 ) 2 × 6 + ( 5 − 5.33 ) 2 × 4 + ( 6 − 5.33 ) 2 × 5 + ( 7 − 5.33 ) 2 × 6 + ( 8 − 5.33 ) 2 × 2 = 27 2.4444 × 4 + 1.7689 × 6 + 0.1089 × 4 + 0.4489 × 5 + 2.7889 × 6 + 7.1289 × 2 = 27 9.7776 + 10.6134 + 0.4356 + 2.2445 + 16.7334 + 14.2578 = 27 54.0623 ≈ 2.44 .
Calculating the Standard Deviation Finally, we calculate the standard deviation by taking the square root of the variance: 2.44 ≈ 1.56 meters.
Final Answer The average height of the harvested teak trees is approximately 5.33 meters, and the standard deviation is approximately 1.56 meters. These values give us an idea of the central tendency and spread of the tree heights.
Examples
Understanding the distribution of tree heights can be useful in forestry and timber management. For example, knowing the average height and standard deviation can help estimate the total volume of timber that can be harvested from a forest. This information can also be used to make decisions about when to harvest trees to maximize yield and profitability. Furthermore, this type of statistical analysis can be applied to other areas of agriculture, such as analyzing crop yields or livestock weights, to optimize farming practices and resource allocation.
