Standard Error

Definition

The standard error is a measure of the variability or dispersion of a sample statistic, such as the sample mean. It represents the standard deviation of the sampling distribution of a statistic, providing an estimate of how much the statistic is likely to vary from one sample to another drawn from the same population.

5 Must Know Facts For Your Next Test

1. The standard error is used to quantify the uncertainty associated with a sample statistic, such as the sample mean or sample proportion.
2. The standard error of the mean is inversely proportional to the square root of the sample size, meaning that as the sample size increases, the standard error decreases.
3. The Central Limit Theorem is a crucial concept in understanding the standard error, as it allows us to make inferences about population parameters based on sample statistics.
4. The standard error is a key component in the calculation of confidence intervals, which provide a range of values likely to contain the true population parameter.
5. Hypothesis testing, which involves comparing a sample statistic to a hypothesized population parameter, also relies on the standard error to determine the statistical significance of the observed difference.

Review Questions

• Explain how the standard error is related to the concept of a sampling experiment (1.6) and how it can be used to make inferences about a population parameter.
  • In a sampling experiment, the standard error represents the variability of a sample statistic, such as the sample mean, around the true population parameter. As the sample size increases, the standard error decreases, and the sampling distribution of the statistic approaches a normal distribution, as described by the Central Limit Theorem (7.1, 7.3, 7.4, 7.5). This allows us to use the standard error to construct confidence intervals (8.4, 8.5, 8.6) and perform hypothesis tests (9.6) to make inferences about the population parameter, even when the population distribution is unknown.
• Discuss how the standard error is used in the context of the Central Limit Theorem (7.1, 7.3, 7.4, 7.5) to make inferences about a population mean, and how this differs when using the normal distribution (8.1) versus the Student's t-distribution (8.2).
  • The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean will approach a normal distribution, with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. This standard deviation is known as the standard error of the mean. When the population standard deviation is known, the standard error can be used to construct a confidence interval for the population mean using the normal distribution (8.1). However, when the population standard deviation is unknown, the standard error is estimated using the sample standard deviation, and the Student's t-distribution is used instead (8.2).
• Explain how the standard error is used in the context of estimating a population proportion (8.3) and how it differs when comparing two independent population proportions (10.3).
  • When estimating a single population proportion, the standard error of the sample proportion is calculated as $\sqrt{\frac{p(1-p)}{n}}$, where $p$ is the sample proportion and $n$ is the sample size. This standard error is then used to construct a confidence interval for the population proportion (8.3). When comparing two independent population proportions (10.3), the standard error is calculated as $\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$, where $p_1$ and $p_2$ are the sample proportions, and $n_1$ and $n_2$ are the respective sample sizes. This standard error is used to test hypotheses and construct confidence intervals for the difference between the two population proportions.

Related terms

Sampling Distribution: The probability distribution of a statistic, such as the sample mean or sample proportion, obtained by repeatedly drawing samples from a population and calculating the statistic for each sample.

Central Limit Theorem: A fundamental theorem in statistics that states that the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution, as the sample size increases.

Confidence Interval: A range of values, calculated from sample data, that is likely to contain an unknown population parameter with a specified level of confidence.