{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Lecture 5: Summaries of Center\n", "\n", "STATS 60 / STATS 160 / PSYCH 10\n", "\n", "**Announcements**\n", "\n", "- Section 05: 4:30 - 5:20pm Hewlett 101 (new location).\n", "- The slides for discussion 2 are online.\n", "\n", "\n", "\n", "## Recap\n", "\n", "**Lecture 4**\n", "\n", "- Data definitions (observational unit, variables, distribution).\n", "- Different types of visualizations (pie charts, bar charts, and line charts).\n", "\n", "\n", "**Today**\n", "\n", "- Data visualizations for quantitative data (histograms and scatter plots).\n", "- Data visualization that use maps (dot maps and cloropleths).\n", "- Summaries of center (mean, median and mode).\n", "\n", " \n", "\n", "# Quantitative Variables\n", "\n", "## Quantitative Data\n", "\n", "
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\n", "\n", "Let's take another look at the old faithful dataset.\n", "\n", "
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\n", "\n", "\n", "## Dot plots for quantitative data\n", "\n", "\n", "\n", "
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\n", "\n", "How could we improve this visualization?\n", "\n", "\n", "## Histograms\n", "\n", "A **histogram** is a more appropriate visualization for a quantitative variable.\n", "First, values are sorted into _bins_, and the number of values in each bin is\n", "plotted as a bar.\n", "\n", "
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\n", "\n", "\n", "\n", "## A Histogram is Not a Bar Chart!\n", "\n", "How is a histogram different from a bar chart?\n", "\n", "
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\n", "\n", "## Histograms vs bar charts\n", "\n", "- Histograms are for quantitative variables and bar chart are for categorical variables.\n", "- The y-axis of a bar chart can be something other than a count. \n", "- For a histogram the y-axis is always a count or a frequency.\n", "\n", "\n", "\n", "## Relationships between Variables\n", "\n", "We can also make a histogram of the `eruption` time of each eruption.\n", "\n", "
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\n", "\n", "\n", "\n", "But how do we understand the relationship between two quantitative variables?\n", "\n", "\n", "## Scatter plots\n", "\n", "In a **scatter plot**, each observation is represented by a point $(x, y)$. \n", "The $x$-coordinate represents the value of one variable, while the \n", "$y$-coordinate represents the value of the other.\n", "\n", "
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\n", "\n", "\n", "\n", "\n", "\n", "\n", "# Maps\n", "\n", "## Dot maps\n", "\n", "
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Coffee Place Geography in 2014 from Flowing Data.
\n", "\n", "## John Snow's Cholera map\n", "\n", "
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John Snow's Cholera map of an 1854 Cholera outbreak.
\n", "\n", "## A problem with dot maps\n", "\n", "
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Dot map of public libraries
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Dot map of gun homicides in 2015 source
\n", "\n", "\n", "\n", "- Be careful of dot maps that are really just population maps!\n", "\n", "## Cloropleths\n", "\n", "\n", "- A *cloropleth* is a map where regions are colored according to the values of a variable. \n", "- \"cloro\" + \"pleth\" means \"region\" + \"many\".\n", "\n", "\n", "
\"Cloropleths
Cloropleths are often used for election results (source).
\n", "\n", "## Cloropleths\n", "\n", "
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A cloropleth showing population density from Our World in Data.
\n", "\n", "## Problems with cloropleths\n", "\n", "
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\n", "\n", "\u201cOh I love those beautiful red areas, that middle of the map. There\u2019s just a little blue here, and a little blue, everything else is bright red.\u201d\n", "\u2014Donald Trump\n", "\n", "
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\n", "\n", "\n", "What is misleading about this chloropleth showing the 2016 election results?\n", "\n", "## Alternatives\n", "\n", "
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A dot map to show population density (source).
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A cartogram to show number of electoral votes from Wikipedia.
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\n", "\n", "## Visualization summary\n", "\n", "- When making a visualization, think about the **number of variables** and the **type of variable** (quantitative or categorical).\n", "- For a **single** variable:\n", " - **Categorical**: bar chart or pie chart.\n", " - **Quantitative**: histogram.\n", "- For **multiple** variables:\n", " - **Two categorical**: stacked bar chart.\n", " - **Two quantitative**: scatter plot.\n", " - **One quantitative, one categorical**: side-by-side histograms.\n", "- For a variable that changes **over time**: line chart.\n", "- For a variable that changes **over locations**: dot map or chloropelth (maps).\n", "\n", "\n", "# Summaries of center\n", "\n", "## USA Women's Eight Rowing\n", "\n", "Shown below are stats for the members of the USA Women's Eight rowing team \n", "that competed at the 2024 Paris Olympics.\n", "\n", "
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\n", "\n", "## USA Women's Eight Rowing\n", "\n", "\n", "- In the dataset of rower weights:\n", "\n", " a. What are the observational units?\n", " b. Is weight a quantitative or categorical variable? What visualization could we use to represent the variable weight?\n", "\n", "\n", "- The rowers are the observational units.\n", "- Weight is a quantitative variable. We could use a histogram.\n", "\n", "## Histogram of weights\n", "\n", "\n", "\n", "
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\n", "\n", "\n", "\n", "But what if we wanted to summarize the data by a single number?\n", "\n", "\n", "## Mean\n", "\n", "One common summary of a quantitative variable is the **mean** \n", "(or **average**, although this is less precise).\n", "\n", "\n", "To calculate the mean, add up the numbers and divide \n", "by how many there are:\n", "$$\n", " \\bar x = \\text{mean} = \\frac{x_1 + x_2 + \\dots + x_n}{n}.\n", "$$\n", "\n", "- $n$ is the number of observational units.\n", "- $x_1,x_2,\\ldots,x_n$ are the different values of a variable.\n", "- $\\bar x$ is a common shorthand for the mean of the values $x_1,\\ldots,x_n$.\n", "\n", "## Mean of the rowers\n", "\n", "Calculate the mean weight of the rowers.\n", "\n", "\n", "$$\n", " \\text{mean} = \\frac{170 + 180 + 115 + 170 + 175 + 170 + 180 + 180 + 160}{9} \\approx 166.7.\n", "$$\n", "\n", "\n", "Surprisingly the mean was not used to summarize data until about 1720! See [this article](https://web.stanford.edu/class/datasci112/readings/mean.pdf) for more history.\n", "\n", "\n", "## Interpreting the Mean\n", "\n", "The mean $\\bar x \\approx 166.7$ measures the \"center\" of the distribution.\n", "\n", "
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\n", "\n", "\n", "\n", "It is where the histogram would \"balance\" if we put it on a scale.\n", "\n", "\n", "## Median\n", "\n", "The mean is not the only way to summarize the center of a distribution.\n", "Another summary is the **median**, the middle value when the data is sorted \n", "in order.\n", "\n", "\n", "Calculate the median weight of the rowers.\n", "\n", "$$\n", " 115, 160, 170, 170, 170, 175, 180, 180, 180\n", "$$\n", "\n", "\n", "$$\n", " 115, 160, 170, 170, \\underbrace{170}_{\\text{median}}, 175, 180, 180, 180\n", "$$\n", "\n", "## The median \n", "\n", "When $n$ is even, there are two middle numbers. The median is the mean of the\n", "two middle numbers.\n", "\n", "\n", "\n", "Calculate the median weight of the $n=8$ rowers, excluding the coxswain.\n", "\n", "$$\n", " 160, 170, 170, \\underbrace{170, 175}_{\\text{median} = 172.5}, 180, 180, 180\n", "$$\n", "\n", "\n", "## Interpreting the Median\n", "\n", "The median $170$ is another summary of the \"center\" of the distribution.\n", "\n", "\n", "
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\n", "\n", "\n", "\n", "It is the value where half the data is below and half the data is above.\n", "\n", "## Mode\n", "\n", "The **mode** is another way of measuring the center of a distribution. The mode is value that appears most often.\n", "\n", "The weights of the rowers have two modes:\n", "\n", "\n", "$$\n", " 115, 160, \\underbrace{170, 170, 170}_{\\text{mode 1}}, 175, \\underbrace{180, 180, 180}_{\\text{mode 2}}\n", "$$\n", "\n", "\n", "This is an example of a _bimodal_ distribution.\n", "\n", "## The mode\n", "\n", "Modes are peaks in the histogram.\n", "\n", "
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\n", "\n", "## Mean vs. Median vs. Mode\n", "\n", "We have now seen three different summaries of center:\n", "\n", "- $\\displaystyle \\text{mean} = \\frac{170 + 180 + 115 + 170 + 175 + 170 + 180 + 180 + 160}{9}\\approx 166.7$\n", "- $\\displaystyle 115, 160, 170, 170, \\underbrace{170}_{\\text{median}}, 175, 180, 180, 180$\n", "- $\\displaystyle 115, 160, \\underbrace{170, 170, 170}_{\\text{mode 1}}, 175, \\underbrace{180, 180, 180}_{\\text{mode 2}}$\n", "\n", "\n", "What would happen to the mean, median and mode, if the coxswain weighed only 90 pounds? \n", "What if the coxswain weighed 140 pounds?\n", "\n", "\n", "_Answer:_ The mean would change, but the median and mode would not.\n", "\n", "\n", "**Moral:** The mean is sensitive to outliers (in either direction), but the \n", "median and mode are not. Statisticians say that the median is more \"robust\" than the\n", "mean.\n", "\n", "## Sensitivity of the mean\n", "\n", "- Recall the general formula for the mean:\n", "\n", "$$ \n", "\\bar x = \\text{mean} = \\frac{x_1 + x_2 + \\dots + x_n}{n}.\n", "$$\n", "\n", "- Changing a single data point will change the mean by a proportional amount. \n", "\n", "\n", "- On the other hand, there is a limit to how much any single data point can change the median or the mode.\n", "\n", "\n", "\n", "## Exercise\n", "\n", "Shown below is a histogram of the arrival delays from the flights data.\n", "\n", "
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\n", "\n", "How do you think the mean and median of the arrival delays compare?\n", "\n", "\n", "- The mean will be bigger than the median.\n", "- The mean is around 7 minutes and the median is -5 minutes.\n", "\n", "\n", "\n", "## The Center Doesn't Tell the Whole Story\n", "\n", "Many people think that the mean/median represent the \"typical\" value, \n", "but this is not always the case.\n", "\n", "\n", "\n", "\n", "
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\n", "\n", "\n", "Consider the Old Faithful eruption times.\n", "\n", "
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\n", "\n", "\n", "- The mean eruption time is about $3.5$ minutes. \n", "\n", "- If we only reported this number, we would miss the fact that most \n", "eruptions are either much shorter or much longer! \n", "\n", "- This is another _bimodal_ distribution.\n", "\n", "\n", "
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\n", "\n", "\n", "## Variability\n", "\n", "Shown below are histograms of daily average temperatures in two cities.\n", "\n", "Chicago\n", "\n", "
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\n", "\n", "\n", "\n", "The means of the two cities are about the same ($53.25^\\circ\\text{F}$ for Chicago \n", "vs. $53.07^\\circ\\text{F}$ for Seattle), but the distributions are very \n", "different.\n", "\n", "\n", "## Recap\n", "\n", "- The mean, the median and the mode are three summaries of center.\n", "- The mean is sensitive to outliers.\n", "- However, summaries of center don't paint the full picture.\n", "\n", "\n", "## Looking ahead\n", "\n", "**Tomorrow**\n", "\n", "- Review and solution to practice quiz 1.\n", "- Vibe coding to make visualizations.\n", "\n", "**Friday**\n", "\n", "- Misleading data visualizations.\n", "- Quiz on data visualizations and summaries of center.\n" ] } ], "metadata": { "kernelspec": { "display_name": "Phython (JB)", "language": "python", "name": "jb-python" } }, "nbformat": 4, "nbformat_minor": 4 }