Corn plants from a test plot are measured, and the distribution of heights at 10-cm intervals is recorded in the following table:  
 Height (cm)      Plants (no.)       
100                    20       
110                    60       
120                    90       
130                   130       
140                   180      
 150                   120       
160                    70        
170                   50        
180                   40 
Calculate 
(a) the mean height, 
(b) the variance, 
(c) the standard deviation, and 
(d) the standard error of the mean. 
Plot a rough graph of plant height against frequency. Do the values represent a normal distribution? Based on your calculations, how would you assess the variation within this population?

Verified step by step guidance

1. Calculate the mean height: Multiply each height by the number of plants at that height, sum these products, and divide by the total number of plants.
2. Calculate the variance: For each height, subtract the mean height, square the result, multiply by the number of plants at that height, sum these values, and divide by the total number of plants.
3. Calculate the standard deviation: Take the square root of the variance.
4. Calculate the standard error of the mean: Divide the standard deviation by the square root of the total number of plants.
5. Plot a rough graph: Use the height intervals on the x-axis and the number of plants on the y-axis to create a histogram. Assess normality by checking if the graph resembles a bell curve.

Key Concepts

Mean

The mean is the average value of a dataset, calculated by summing all the values and dividing by the number of observations. In this context, it represents the average height of the corn plants, providing a central tendency measure that helps summarize the data. Understanding how to compute the mean is essential for analyzing the overall growth of the plants.

Variance and Standard Deviation

Variance measures the spread of a set of values around the mean, indicating how much the heights of the corn plants differ from the average height. The standard deviation, the square root of variance, provides a more interpretable measure of this spread in the same units as the data. Both concepts are crucial for assessing the variability within the population of corn plants.

Normal Distribution

A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this question, determining whether the plant height data follows a normal distribution is important for understanding the population's characteristics and making inferences about the overall growth patterns.