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Since this variable is quantitative, it is automatically ordinal 0 3. These data are referred to as the observations in the experiment. Example 3. Even though fold changes are sometimes reported as integers, intermediate values are possible and can be determined with more precision in the measurements. Thus, "expression" of a gene is a random variable. A description such as "gene XYZ underwent a 2. When a random variable such as gene expression or height of a plant is observed repeatedly on several subjects different genes, plants, etc.

For quantitative variables, the most commonly used graphical display methods are histograms, dot plots, scatter plots, and box plots, described in Section 3. Numerical Ways to Describe Data vVhen several observations of the same type are obtained e.

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This can be done in different ways. The tables contain the labels or possible categories and COUNTS of how often these labels were observed in the experiment. Some mice are left unvaccinated to function as a control. All mice are infected with the yellow fever virus and, after an appropriate incubation period, live and dead mice are counted.

One vanable describes the type of vaccine the mouse received A, B, or none , and the other variable states whether the mouse is alive or dead.

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Consequently, the median can also be thought of as ,the 50t 1 percentile. In Excel: To compnt.

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If n is even, then the median s the average of the two middle observations. Unlike the variance, which has nonsensical units, however, its units are the same as the original observation measurements. Commonly used 2 2 symbols for standard deviation variance are s, CY s , CY.

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Standard deviation should not be confused with standard error Section 3. To compute a standard deviation, click on any em. The range can be used as a crude measure of spread in the data, but it is more susceptible than the variance standard deviation to misrepresent the true variation in the data if there are uncharacteristically large or small observations among Range the data points. It is the distance between the third and first quartiles of the data. Outliers can be caused by errors during the measurement process or during the recording of data.

Outliers may also be measurements that differ from the majority of the data points for legitimate reasons e. The decision as to whether or not an observation is an outlier is a subjective one. A statistical rule of thumb to decide whether or not an observation may be considered an outlier uses the IQR of the data. Using all data points including possibly suspected outliers , compute the first and third quartiles Ql and Q3 as well as the interquartile range IQR Q3 - Ql for the data.

If an observation is suspected to be an outlier, double-check the recording of the observation to rule out typographical errors. If the measurement cannot be repeated, a statistical analysis should be performed with and without the outlier.

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If the data point is omitted in the subsequent analysis, then this should be accompanied by an explanation of why this value is considered to be an outlier and what may have caused it to be so different from the bulk of the observations. Exarnple 3. In this experiment, spinach was determined to contain 35 mg of iron per g of spinach. This error gave rise to a 17 Numerical Ways to Describe Data. Department of Agriclllture What about th: valu: for beef?

Computing the quartiles for the above 15 data pomts y1elds Q 0.

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This makes sense. The iron value for beef 6. If the variability in the data is small, then thi; means that the observations are all grouped closely together. If, on the other hand the observations cover a wide range of values, then the variation measure will be large. Similar to the case of center measure, the variance and standard deviation are better suited in cases where there are no extreme outliers among the observations.

In an experiment, progeny of a certain parental cross of a flowering plant were obtained and categorized by their flower color. The results may be listed in table form. Figure 1 displays the data in the form of two bar plots. The numbers of observations that fall into Color Count red white pink 94 37 75 When displaying categorical data, an alternative to the bar graph is a pie chart compare Figs.

The categories are then represented by "slices of pie'' of varying thicknesses rather than by bars. Together, the slices add up to the "whole pie" Fig. The lengths of the bars represents the frequency for each category. Since the values of a categorical variable may not be ordinal, the order of the bars each labeled by the it represents can be altered without changing the meaning of the plot.

How to Choose a Descriptive Measure The mean and median are numerical measures that are used to describe the center of a distribution or to find a "typical" value for a given set of observations. If there are some atypical values outliers among the observations, then the median is a more reliable measure of center than the mean. On the other hand, if there are no outliers and especially if the number of observations is large, then the mean is the preferred measure of center. If the data are to be used to answer a specific question, then vvorking with means rather than medians will make subsequent statistical analysis much more straightforward.

Observed flower color of plants plotted as a bar plot. The order of the categories is arbitrary.

The frequencies of observations i. The bin width and number of bins that should be used in a histogram depend on the data. Histograms may not be appropriate if the number of observations is small. The larger the number of observations, the narrower the bins can be chosen while still accurately portraying the data.

Well-constructed histograms allow a quick overview of the data to check for center, symmetry, outliers, and general shape. Guy studied the duration of life among the male English gentry Guy He recorded the life spans of adults ages 21 and up from pedigrees of country families and from mural tablets. His results are displayed in Figure 3 in the form of a histogram.

Since the number of observations in this example is very large, the histogram bin width can be relatively small e. If the bin width is larger e. The size of the bin width should be determined by the level of detail needed in the study's conclusions. Pie chart of the flower color data from Example 3.

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You can values to use for the bins. Life span in years of the male English gentry. The same data from Guy are displayed with different bin widths in the form of a histogram. In the histogram on the left, the bin width is 10 years and in the histogram on the right, the bin width is 5 years. Two dot plots of the same ten observations plotted horizontally left and vertically right.

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Repeated observations of the same value are plotted above or beside each other. In it, one variable is plot:d on the horizontal x-axis and the other variable is plotted on 1e vertical y-axis. Scatter plots allow visual overview of center and fead for each variable while at the same time giving important ues to possible relationships between the two variables. Observations on the same should appear in the same row. XY and highlight your observation columns.

In this ray, they summarize the data and are thus suited to describe outomes of experiments in which the number of observations is too "Lrge to report each outcome separately as is done in a dot plot.

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Co construct a box plot, compute the minimum, maximum, and melian as well as the first and third quartiles for each data set that rou want to display. Usually, box plots are drawn with the measurenents on a vertical scale, but they could be drawn horizontally as Nell. The box spans from the first to the third quartile of the observations and "tails" or "whiskers" extend to the maximum and minimum observations, respectively.

G data set Guy which contains observations on the life span of the English upper class in the 19th century. Do sovereigns live longer than aristocrats? To be able to make this comparison in a graph, we draw a side-by-side box plot for the available observations on the three groups.

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Overall, sovereigns seem to live a little less long than the members of the other two groups. In Example 3.