6.13
標本分布
各サンプルの平均、割合、標準偏差などの測定された特性を持つ、特定の母集団からのサイズ n の単純なランダムサンプルが与えられた場合、測定されたすべての特性の確率分布は標本分布と呼ばれます。統計がサンプル間でどの程度異なるかは、統計のサンプリング変動として知られています。通常、統計の標本変動は標準誤差…
10個のピッカーホイールを回転させ、結果の平均を見つけることを検討してください。このプロセスは、たとえば20,000回繰り返されます。
プロセスの繰り返しごとに得られたサンプル平均がプロットされ、正規分布グラフのように見えます。
サンプルサイズが大きい場合、分布は正規分布に近くなり、サンプル平均の平均は母平均に近づきます。
平均、分散、サンプル比率などの統計量の値のこのような分布は、サンプリング分布と呼ばれます。
平均と同様に、各サンプルの分散を取得し、右に歪んで見える周波数分布をプロットできます。
この場合でも、サンプルサイズが大きい場合、サンプル分散の平均は母分散に近くなります。
各サンプルの奇数の割合を考慮してグラフをプロットすると、分布はほぼ正規分布パターンに従います。
平均と分散と同様に、サンプルサイズが大きい場合、サンプル比率の平均は母比率に近くなります。
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Q1: What is a sampling distribution and how is it created?
A sampling distribution is the probability distribution of a statistic, such as the mean, variance, or proportion, calculated from multiple simple random samples of the same size from a population. It is created by repeatedly drawing samples, computing the statistic for each sample, and plotting the frequency distribution of those statistics.
Q2: How does sample size affect the shape of a sampling distribution?
As sample size increases, the sampling distribution approaches a normal distribution shape, regardless of the original population distribution. This convergence means that larger samples produce more reliable estimates, with the mean of sample statistics becoming increasingly close to the true population parameter.
Q3: What is sampling variability and how is it measured?
Sampling variability refers to how much a statistic varies from one sample to another. It is measured using the standard error, which is the standard deviation of the sampling distribution. The standard error of the mean is a common example that quantifies the variability of sample means around the population mean.
Q4: Why does the mean of sample means approach the population mean?
When sample size is large, the mean of sample means converges to the population mean due to the law of large numbers. This property holds for other statistics as well: the mean of sample variances approaches the population variance, and the mean of sample proportions approaches the population proportion.
Q5: Can sampling distributions be used for different types of statistics?
Yes, sampling distributions can be constructed for any statistic, including the mean, variance, and proportion. Each statistic has its own sampling distribution with distinct characteristics. For example, the distribution of sample proportions typically follows an approximately normal pattern when sample size is sufficiently large.
Q6: What role does standard error play in statistical inference?
Standard error measures the precision of a sample statistic as an estimate of the population parameter. A smaller standard error indicates that sample statistics cluster more tightly around the true population value, making the estimate more reliable for drawing conclusions about the population.
Q7: How do sampling distributions relate to probability distributions?
A sampling distribution is a specific type of probability distribution that describes the likelihood of different values for a sample statistic. While probability distributions characterize outcomes of random variables, sampling distributions characterize the behavior of statistics computed from repeated samples.