CMSE11624 Statistics for Analytics
Session 2: Descriptive and Summary Statistics
Data Process
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Summarising Data in Graph
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Sample vs Population
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For example, we might want to know the mean weight of a certain species of turtle that has a total population of 800.
Measures of Location: Mean
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Measures of Location: Mode
The mode of a data set is the value that occurs with the (local) greatest frequency.
The greatest frequency can occur at two or more different values.
If the data have exactly two modes, the data are bimodal.
If the data have more than two modes, the data are multimodal.
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Measures of Location: Median
The median of a data set is the value in the middle when the data items are arranged in ascending order.
Whenever a data set has extreme values, the median is the preferred measure of central location.
The median is the measure of location most often reported for annual income and property value data.
A few extremely large incomes or property values can inflate the mean.
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Measures of Location: Median
Measures of Location: Percentiles
A percentile provides information about how the data are spread over the interval from the smallest value to the largest value.
The \(p\)th percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - \(p\)) percent of the items take on this value or more.
Measures of Variability: Variance & Standard Deviation
The variance is the average of the squared differences between each data value and the mean. Standard Deviation is the square root of variance.
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image The sample mean is an unbiased estimator of the population mean.
The sample variance SHOULD BE an unbiased estimator of the population variance. \[\mathbb{E}(s^2) = \sigma^2\]
Measures of Variability: Variance & Standard Deviation
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Measures of Variability: Range & IQR
Range = Largest - Smallest
IQR = Q3 - Q1
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Measures of Both: Skewness
The skewness (3rd moment) is a measure of the asymmetry of the distribution. The skewness \(\gamma_1\) of a random variable X is the third standardised moment \(\tilde {\mu }_{3}\), defined as \[\gamma_1 : \tilde{\mu_3} = \mathbb{E}[\frac{X - \mu}{\sigma}] = \frac{\mu_3}{\sigma^3} = \frac{\mathbb{E}[(X- \mu)^3]}{(\mathbb{E}[(X-\mu)^2])^{3/2}}\]
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Positive skewed = right skewed
Negative skewed = left skewed
Measures of Both: Skewness
In risk measurement, it tells us whether the probability of winning is similar to the probability of losing and the nature of losses (assuming the distributions below are loss distribution ).
Negative skewness means that there is a substantial probability of a big negative return.
Positive skewness means that there is a greater-than-normal probability of a big positive return.
Simpler definition (Pearson median skewness or second skewness coefficient) \[\gamma_2 = \frac{3 (mean - median)}{\sigma}\]
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Example: Hudson Auto Repair
The manager of Hudson Auto would like to have a better understanding of the cost of parts used in the engine tune-ups performed in the shop. She examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest £:
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| 91 | 78 | 93 | 57 | 75 | 52 | 99 | 80 | 97 | 62 |
| 71 | 69 | 72 | 89 | 66 | 75 | 79 | 75 | 72 | 76 |
| 104 | 74 | 62 | 68 | 97 | 105 | 77 | 65 | 80 | 109 |
| 85 | 97 | 88 | 68 | 83 | 68 | 71 | 69 | 67 | 74 |
| 62 | 82 | 98 | 101 | 79 | 105 | 79 | 69 | 62 | 73 |
Frequency Distribution
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Frequency Distribution
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Guidelines for Selecting Number of Classes
Use between 5 and 20 classes.
Data sets with a larger number of elements usually require a larger number of classes.
Smaller data sets usually require fewer classes
Guidelines for Selecting Width of Classes
Use classes of equal width
\(Approximate Class Width = \frac{Largest Value - Smallest Value}{Number of Classes}\)
Frequency Distribution
For this dataset, we can calculate, Approximate Class Width \(= (109 - 52)/6 = 9.5 \simeq 10\)
For Hudson Auto Repair, if we choose six classes:
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Relative Frequency and Percentage Frequency Distributions
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Histogram (simple version)
Another common graphical presentation of quantitative data is a histogram.
The variable of interest is placed on the horizontal axis.
A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percentage frequency.
Unlike a bar chart, a histogram has no natural separation between rectangles of adjacent classes.
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Density Plot
The kernel density estimate to smooth the histogram, providing complementary information about the shape of the distribution:
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Density Plot
A density plot is a robust way to get the shape of the distribution.
Bar plot
A standard bar chart/ graph is a chart or graph that presents categorical data with rectangular bars with heights/lengths proportional to the values that they represent.
Bar plot
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A bar plot represents an estimate of the central tendency for a numeric variable with the height of each rectangle and provides some indication of the uncertainty around that estimate using error bars.
Mosaic plot
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A mosaic plot is a visualisation technique suitable for contingency tables that resembles a standardised/percentage stacked bar plot with the benefit that we still see the relative group sizes of the primary variable as well.
Boxplot
A box plot (box-and-whisker plot) shows the distribution of quantitative data in a way that facilitates comparisons between variables or across levels of a categorical variable.
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Positive skewed (mean \(>\) median)
Upper whisker = Q3 + 1.5 * IQR, Lower whisker = Q1 - 1.5 * IQR
Violinplot/ Pirate Plot
A violin plot plays a similar role as a box and whisker plot. It shows the distribution of quantitative data across several levels of one (or more) categorical variables such that those distributions can be compared.
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Components correspond to actual datapoints
Kernel density estimation
