1.7 Summary Statistics for One Quantitative Variable

Calculate measures of center and position for quantitative data.

• Two commonly used measures of center in the distribution of a quantitative variable are the mean and median.
• The mean is the sum of all the values divided by the number of values and can be found with and without using technology. For a sample, the mean is denoted by x-bar: 1xx= ∑n i i=1 n , where xi represents the ith data point in the sample and n represents the number of data values in the sample.
• The median is the middle value when the data set is ordered from smallest to largest and can be found with and without using technology. One common method for determining the median of a data set with an even number of values is to use the mean of the two middle values. A common method for determining the median of a data set with an odd number of values is to use the value in the middle of all the values.
• In an ordered data set, the smallest value is the minimum value, and the largest value is the maximum value.
• The first quartile, denoted by Q1, is the median value of the lower half of the ordered data set from the minimum value to the position of the median. Approximately 25% of the values in the data set are less than or equal to Q1. The third quartile, denoted by Q3, is the median value of the upper half of the ordered data set from the position of the median to the maximum value. Approximately 75% of the values in the data set are less than or equal to Q3. The second quartile, Q2, is also the median of the data set. Q1 and Q3 form the boundaries for the middle 50% of values in an ordered data set.
• The pth percentile is the value that has p% of the data less than or equal to it when the data set is ordered from smallest to largest. The first and third quartiles are the 25th and 75th percentiles, respectively.

Calculate measures of variability for quantitative data.

• Three commonly used measures of variability (or spread) in the distribution of a quantitative variable are the range, interquartile range, and standard deviation.
• The range is the difference between the maximum data value and the minimum data value.
• The interquartile range (IQR) is the difference between the third and first quartiles: Q3 − 1.Q
• The standard deviation is a typical deviation of the data values from their mean and can be found with and without using technology. The sample standard deviation is denoted by s and calculated by 1s = ()xx 2 n 1 i −− ∑ , where xi is the data value, x is the mean, and n is the number of data values in the sample. The square of the sample standard deviation, s2, is called the sample variance.

Calculate different units of measurement for summary statistics.

• Changing units of measurement affects the values of the calculated statistics.

Calculate outliers for quantitative data.

• There are many methods for determining potential outliers. Two methods frequently used are as follows:
  • i. An outlier is a value located more than 1.5I× QR above the third quartile or more than 1.5I× QR below the first quartile.
  • ii. An outlier is a value located more than 2 standard deviations above, or below, the mean.

Compare multiple quantitative one-variable summary statistics.

• Summary statistics can be used to compare features of two or more independent samples, including center, variability, shape, and outliers.

Justify the selection of a summary statistic for describing quantitative data.

• The median and IQR are considered a resistant (or robust) measure of center and measure of variability, respectively, because outliers do not greatly (if at all) affect their values. Because outliers can affect their values greatly, the mean is considered a nonresistant (or non-robust) measure of center, and the range and standard deviation are considered nonresistant (or nonrobust) measures of variability.
• Summary statistics of a quantitative variable may reveal information that can be used to justify claims about the variable in context.