Question

An agronomist measured the heights of \(n\) corn plants. \({ }^{5}\) The mean height was \(220 \mathrm{~cm}\) and the standard deviation was \(15 \mathrm{~cm}\). Calculate the standard error of the mean if (a) \(n=25\) (b) \(n=100\)

Short answer

The standard error of the mean is 3 cm for \(n=25\) and 1.5 cm for \(n=100\).

Step-by-step solution

Understanding Standard Error

The standard error of the mean (SEM) is a measure of how much the sample mean (average) you have might vary from the true population mean. The SEM is calculated by dividing the standard deviation by the square root of the sample size (n). The formula for SEM is: \[ SEM = \frac{\sigma}{\sqrt{n}} \] where \( \sigma \) is the standard deviation and \( n \) is the sample size.

Calculating Standard Error for 25 Samples

Plug the given values into the SEM formula for \( n = 25 \). Given that the standard deviation (\( \sigma \)) is 15 cm, we get: \[ SEM = \frac{15}{\sqrt{25}} = \frac{15}{5} = 3 \text{ cm} \]

Calculating Standard Error for 100 Samples

Similarly, for \( n = 100 \), we calculate: \[ SEM = \frac{15}{\sqrt{100}} = \frac{15}{10}= 1.5 \text{ cm} \] Thus, the standard error when \( n = 100 \) is 1.5 cm.

Key concepts

Sample Size

The sample size, denoted by the symbol n, plays a vital role in statistical analysis as it influences the precision of your estimated measurements, such as the mean height of corn plants in our example. A larger sample size typically yields a smaller Standard Error of the Mean (SEM), which indicates that the mean of your sample is a more precise estimate of the true population mean.

This is because with more data, there is less fluctuation and more stability in your results. You can think of it like trying to guess the average height of people in a city. If you only ask a handful of individuals, your average might be far off. But if you ask hundreds, you're likely to get a much more accurate average height.

In the agronomist's case, increasing the sample size from 25 to 100 corn plants led to a decrease in the SEM. With fewer samples, there's a greater chance of random variation affecting the mean, therefore a larger sample size is always more desirable to limit this random variability and to get closer to the true population mean.

Standard Deviation

Standard deviation, symbolized by σ, is a measure of the amount of variation or dispersion within a set of values. Think of it as an average distance from the mean. In our textbook problem, the standard deviation of the corn plants' height is 15 cm.

This means that, on average, the height of each corn plant varies by about 15 cm from the mean height of 220 cm. A smaller standard deviation indicates that the heights of the corn plants are closer to the mean (less spread out), whereas a larger standard deviation indicates more variation in plant heights.

When calculating SEM, the standard deviation is used as a key component since it provides information on how spread out the measurements are. It's necessary to understand that the SEM denotes how the mean of samples might spread if you were to draw multiple samples from the same population, whereas standard deviation indicates the spread of individual values within a single sample.

Population Mean

Population mean, often denoted by the Greek letter μ, is the average of all the measurements in the full population. In statistics, it’s the value that is being estimated by using the sample mean. The term 'population' can be misleading; it doesn’t necessarily mean people. It could refer to anything that's being measured, such as the height of corn plants.

In the agronomist's study, the mean height of the sample of corn plants is 220 cm. While this number provides an estimate, the population mean is the average height of all such plants, in this case theoretically, under similar conditions. It is what we're trying to infer or estimate using our sample. Remember that a sample mean can vary from the population mean due to sampling error, which the SEM helps to quantify.

Statistical Variability

Statistical variability, or statistical variation, refers to the natural variation that occurs in data. No two corn plants are exactly alike, hence their heights will naturally vary; this is variability. It is an intrinsic feature of any set of data that involves measurements of various subjects or objects.

Variability can be quantified using various statistics, with standard deviation being a particularly important one because it directly influences the SEM. Lower variability within a population results in a smaller standard deviation and, consequently, a smaller SEM. Factors affecting variability include genetic differences, environmental factors, and measurement errors.

It is crucial to realize that variability is not an error but a reality of data collection. Mitigating variability's effect in conclusions about a population is where sample size and SEM become extremely useful for agronomists and statisticians alike.