Observed sample statistics for Randomization Test for 1-Sample Mean

Find definitions and interpretation guidance for every observed sample statistic that is provided with randomization test for 1-sample mean.

N

The sample size (N) is the total number of observations in the original sample. Minitab takes resamples of this sample size to form the bootstrap samples.

Mean

The mean is the average of the data, which is the sum of all the observations divided by the number of observations.

For example, the wait times (in minutes) of five customers in a bank are: 3, 2, 4, 1, and 2. The mean waiting time is calculated as follows:
On average, a customer waits 2.4 minutes for service at the bank.

Interpretation

Minitab displays two different mean values, the mean of the observed sample and the mean of the bootstrap distribution. The mean of the observed sample is an estimate of the population mean. The mean of the bootstrap distribution is usually close to the hypothesized mean. The larger the difference between these two values, the more evidence you would expect against the null hypothesis.

StDev (observed sample)

The standard deviation is the most common measure of dispersion, or how spread out the data are about the mean. The symbol σ (sigma) is often used to represent the standard deviation of a population, while s is used to represent the standard deviation of a sample. Variation that is random or natural to a process is often referred to as noise. Because the standard deviation is in the same units as the data, it is usually easier to interpret than the variance.

The standard deviation of the observed sample is an estimate of the standard deviation of the population.

Interpretation

Use the standard deviation to determine how spread out the data are from the mean. A higher standard deviation value indicates greater spread in the data. A good rule of thumb for a normal distribution is that approximately 68% of the values fall within one standard deviation of the mean, 95% of the values fall within two standard deviations, and 99.7% of the values fall within three standard deviations. The standard deviation of the sample data is an estimate of the population standard deviation.

The standard deviation can also be used to establish a benchmark for estimating the overall variation of a process.
Hospital 1
Hospital 2
Hospital discharge times

Administrators track the discharge time for patients who are treated in the emergency departments of two hospitals. Although the average discharge times are about the same (35 minutes), the standard deviations are significantly different. The standard deviation for hospital 1 is about 6. On average, a patient's discharge time deviates from the mean (dashed line) by about 6 minutes. The standard deviation for hospital 2 is about 20. On average, a patient's discharge time deviates from the mean (dashed line) by about 20 minutes.

Variance

The variance measures how spread out the data are about their mean. The variance is equal to the standard deviation squared.

Interpretation

The greater the variance, the greater the spread in the data.

Because variance (σ2) is a squared quantity, its units are also squared, which may make the variance difficult to use in practice. The standard deviation can be easier to use because it is a more intuitive measurement. For example, a sample of waiting times at a bus stop may have a mean of 15 minutes and a variance of 9 minutes 2. Because the variance is not in the same units as the data, the variance is often displayed with its square root, the standard deviation. A variance of 9 minutes2 is equivalent to a standard deviation of 3 minutes.

Sum

The sum is the total of all the data values. The sum is also used in statistical calculations, such as the mean and standard deviation.

Minimum

The minimum is the smallest data value.

In these data, the minimum is 7.

13 17 18 19 12 10 7 9 14

Interpretation

Use the minimum to identify a possible outlier or a data-entry error. One of the simplest ways to assess the spread of your data is to compare the minimum and maximum. If the minimum value is very low, even when you consider the center, the spread, and the shape of the data, investigate the cause of the extreme value.

Median

The median is the midpoint of the data set. This midpoint value is the point at which half the observations are above the value and half the observations are below the value. The median is determined by ranking the observations and finding the observation that are at the number [N + 1] / 2 in the ranked order. If the number of observations are even, then the median is the average value of the observations that are ranked at numbers N / 2 and [N / 2] + 1.

For this ordered data, the median is 13. That is, half the values are less than or equal to 13, and half the values are greater than or equal to 13. If you add another observation equal to 20, the median is 13.5, which is the average between the 5th observation (13) and the 6th observation (14).

Interpretation

The median and the mean both measure central tendency. But unusual values, called outliers, affect the median less than they affect the mean. When you have unusual values, you can compare the mean and the median to decide which is the better measure to use. If your data are symmetric, the mean and median are similar.

Maximum

The maximum is the largest data value.

In these data, the maximum is 19.

13 17 18 19 12 10 7 9 14

Interpretation

Use the maximum to identify a possible outlier or a data-entry error. One of the simplest ways to assess the spread of your data is to compare the minimum and maximum. If the maximum value is very high, even when you consider the center, the spread, and the shape of the data, investigate the cause of the extreme value.

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