Let j denote a row vector consisting entirely of 1s. Provethat a nonnegative matrix P is a stochastic matrix ifand only if jP j.

Monday, January 22, 2018 Stats Section 2.1 - Visualizing Variation in Numerical Data Distribution of a sample: organization of data, list that records the values that were observed and the frequencies of these values Frequency: the count of how many times the value occurred in the collection of data - The ﬁrst step to almost every statistical investigation is to visualize the distribution of the sample so that we can see patterns more easily - To examine distributions, we use a two-step process: • See it • Summarize it Two useful methods of for visualizing distributions are dot plot (simpler) and histograms (more common and useful) Dotplot: Put a dot above a number line where each value occurs to get a sense of frequency by seeing how high the dots stack up - Use when data set is small, and lack of tech Histograms Histogram: produce a smoother graphic by grouping observations into intervals, called bins, which are formed by dividing the number line into equal width and then counting how many observations fall into each bin - Rules need to be made: • For when an observation lands exactly on a boundary, just be consistent - Right/left-hand rule to put value on boundary into one bin • Using wider bins hides more details, but don’t make it too small, 3 is good Relative frequency Histograms - Variation on the histogram, change vertical axis from frequency to relative frequency • Relative frequency: a proportion, label is 0-1 - Ex- ﬁrst bin has 5 observations (out of 28), so 5/28=0.18 1