Section 3.6 Summary
Subsection 3.6.1 Glossary
- Median
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A number that separates ordered data into halves: half the values are the same number or smaller than the median and half the values are the same number or larger than the median. The median may or may not be part of the data.
- Quartiles
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The numbers that separate the data into quarters. Quartiles may or may not be part of the data. The second quartile is the median of the data.
- Outlier
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An observation that does not fit the rest of the data.
- Interquartile Range (IQR)
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The distance between the third quartile (Q3) and the first quartile (Q1). IQR = Q3 - Q1.
- Mean
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A number that measures the central tendency. A common name for mean is ’average.’ The term ’mean’ is a shortened form of ’arithmetic mean.’ By definition, the mean for a sample (denoted by \(\bar x\)) is\begin{equation*} \bar x = \frac{\mbox{Sum of all values in the sample}}{\mbox{Number of values in the sample}}\;. \end{equation*}
- (Sample) Variance
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Mean of the squared deviations from the mean. Square of the standard deviation. For a set of data, a deviation can be represented as \(x - \bar x\) where \(x\) is a value of the data and \(\bar x\) is the sample mean. The sample variance is equal to the sum of the squares of the deviations divided by the difference of the sample size and 1:\begin{equation*} s^2 = \frac{\mbox{Sum of the squares of the deviations}}{\mbox{Number of values in the sample}-1}\;. \end{equation*}
- (Sample) Standard Deviation
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A number that is equal to the square root of the variance and measures how far data values are from their mean. \(s = \sqrt{s^2}\text{.}\)
Subsection 3.6.2 Discuss in the forum
An important practice is to check the validity of any data set that you are supposed to analyze in order to detect errors in the data and outlier observations. Recall that outliers are observations with values outside the normal range of values of the rest of the observations.
It is said by some that outliers can help us understand our data better. What is your opinion?
When forming your answer to this question you may give an example of how outliers may provide insight or, else, how they may obstruct our understanding. For example, consider the price of a stock that tend to go up or go down at most 2% within each trading day. A sudden 5% drop in the price of the stock may be an indication to reconsidering our position with respect to this stock.
Subsection 3.6.3 Commonly Used Symbols
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The symbol \(\sum\) means to add or to find the sum.
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\(n\) = the number of data values in a sample.
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\(\bar x\) = the sample mean.
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\(s\) = the sample standard deviation.
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\(f\) = frequency.
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\(f/n\) = relative frequency.
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\(x\) = numerical value.
Subsection 3.6.4 Commonly Used Expressions
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\(x \times (f_x/n)\) = A value multiplied by its respective relative frequency.
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\(\sum_{i=1}^n x_i\) = The sum of the data values.
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\(\sum_x (x \times f_x/n)\)= The sum of values multiplied by their respective relative frequencies.
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\(x - \bar x\) = Deviations from the mean (how far a value is from the mean).
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\((x - \bar x)^2\) = Deviations squared.
Subsection 3.6.5 Formulas
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Mean: \(\bar x = \frac{1}{n} \sum_{i=1}^n x_i = \sum_x \big(x \times (f_x/n)\big)\)
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Variance: \(s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar x)^2 = \frac{n}{n-1}\sum_x \big((x - \bar x)^2\times (f_x/n)\big)\)
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Standard Deviation: \(s = \sqrt{s^2}\)
