2.1 STEM-AND-LEAF GRAPHS (STEMPLOTS), LINE GRAPHS, AND BAR GRAPHS

• Stem-and-leaf graphs (stemplots) overview

  • Purpose: organize data by splitting each observation into a stem and a leaf.

  • Leaves are the final significant digit; stems capture the remaining leading digits.

  • Example conventions:

  • The observation 23 has stem 2 and leaf 3.

  • The observation 432 has stem 43 and leaf 2.

  • Construction steps:

  • Write stems in a vertical line from smallest to largest.

  • Draw a vertical line to the right of the stems.

  • Write leaves in increasing order next to their corresponding stem.

  • Example task: creating a stem-and-leaf graph for a given dataset (as in the text).

• Line graphs

  • Used to display data that change over an ordered category or time (e.g., frequency of reminders per week).

  • Data are typically shown in a table (Table 2.7) and a figure (Figure 2.2).

  • Interpretation focus: trend over time, rise/fall patterns, and comparisons across categories.

• Bar graphs

  • Used to display categorical or grouped numerical data.

  • Data example (Facebook users by age group, end of 2011):

  • Age groups and user counts:

    • 13–25: 65,082,280 users; 45%

    • 26–44: 53,300,200 users; 36%

    • 45–64: 27,885,100 users; 19%

  • Bar height corresponds to the measure shown (counts or proportions).

  • Construction points: order of categories, equal-width bars, clear axis labels, and a legend if needed.

2.2 HISTOGRAMS, FREQUENCY POLYGONS, AND TIME SERIES GRAPHS

• Histograms

  • A histogram consists of contiguous boxes (bars) with both a horizontal and a vertical axis.

  • Horizontal axis: represents the data (e.g., a measurement such as distance).

  • Vertical axis: represents frequency, relative frequency, percent frequency, or probability.

  • The histogram conveys the shape of the data, the center, and the spread, similarly to a stem-and-leaf plot.

  • Construction steps:

  • Decide the number of bars (classes) representing the data.

• Frequency polygons

  • A line graph that connects the midpoints of class intervals (the class marks) to show the distribution shape.

  • Useful for comparing distributions or showing the overall pattern of a frequency distribution.

• Time series graphs

  • Plot values of a variable across time (e.g., Annual CPI over years).

  • Axes typically include Time (years) on the horizontal axis and the measured value on the vertical axis.

  • Interpretation emphasizes trends, cycles, and long-run changes over time.

2.3 MEASURES OF THE LOCATION OF THE DATA

• Median

  • Definition: a number that separates ordered data into halves; the center of the data in the sense of positional rank.

  • Properties:

  • For ordered data, half of the values are ≤ the median and half are ≥ the median.

  • The median need not be an observed value.

  • Example (data):

  • Data: 1, 11.5, 6, 7.2, 4, 8, 9, 10, 6.8, 8.3, 2, 2, 10, 1

  • Ordered: 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5

  • With n = 14 (even), the median is between the 7th and 8th values:
    $ext{median} = rac{6.8 + 7.2}{2} = 7.0$

• Quartiles (Q1, Q2, Q3)

  • Q2 is the median (second quartile).

  • Q1: the median of the lower half of the data.

  • Q3: the median of the upper half of the data.

  • IQR (interquartile range):
    $ext{IQR} = Q<em>3 - Q</em>1$

  • IQR use and outliers:

  • The IQR helps assess the spread of the middle 50% of data.

  • Potential outliers are values below $Q<em>1 - 1.5 imes ext{IQR}$ or above $Q</em>3 + 1.5 imes ext{IQR}$.

• Percentiles

  • A percentile indicates the relative standing of a data value when data are ordered.

  • Interpretation: e.g., the 15th percentile means 15% of data values are ≤ that value.

  • Two percentile concepts in the text:

  • Percentile formula for kth percentile (i.e., position-based):

    • Ordering data from smallest to largest, let $i = rac{k(n+1)}{100}$

    • If $i$ is a positive integer, the kth percentile is the data value at position $i$.

    • If $i$ is not an integer, round $i$ up and down to the nearest integers and average the two data values at those positions.

  • Percentile of a given value within a data set (rank-based percentile):

    • Let $x$ be the position counting from the bottom to the data value of interest, $y$ the number of data values equal to that value, and $n$ the total number of data.

    • Compute the percentile as:
      $ext{Percentile} = rac{x + 0.5y}{n} imes 100 ext{.}$

• Outliers (context)

  • Outliers are data points that stand away from the main body of the data; they may be errors or important insights.

2.4 BOX PLOTS

• Box plots (box-and-whisker plots) provide a graphical summary of data concentration and spread.

• Five-number summary used: minimum, Q1, median, Q3, maximum.

• Construction idea:

  • A box spans from Q1 to Q3 with a line inside for the median.

  • Whiskers extend to the minimum and maximum data values (unless outliers are defined separately).

• Example application: heights of 40 students (data listed in the text) illustrating how a box plot summarizes spread and center visually.

2.5 MEASURES OF THE CENTER OF THE DATA

• The center of a data set is described by measures of location, primarily:

  • Mean (average):

  • For a sample of n values, $\bar{x} = rac{1}{n} extstyle\bigg(<br>\sum<em>{i=1}^n x</em>i\bigg)$

  • For a population, the mean is ar{x} ext{ or } ext{often }oldsymbol{bc} depending on notation, but the key idea is the same: the arithmetic average.

  • Median: discussed in 2.3.

• When to prefer mean vs. median

  • The median is generally more robust to extreme values and outliers; hence it is a better measure of center for skewed data.

  • The mean is the most common measure of center and is used in many statistical formulas and inferential procedures.

• Mode

  • The most frequent value(s) in the data.

  • There can be more than one mode (multimodal). A data set with two modes is called bimodal.

  • The mode can be used for qualitative data as well (e.g., color categories).

• Grouped data: mean of grouped frequency tables

  • When only grouped data are available, the exact mean cannot be computed; estimate via the mean of the frequency table.

  • Key quantities:

  • $f$ = frequency of the interval

  • $m$ = midpoint of the interval

2.6 SKEWNESS AND THE MEAN, MEDIAN, AND MODE

• Symmetrical distribution

  • A distribution is symmetrical if there exists a vertical line around which the left and right sides are mirror images.

  • In a perfectly symmetrical distribution, mean = median = mode.

• Skewed distributions

  • Left-skewed (skewed to the left): tail on the left; mean < median < mode.

  • Right-skewed (skewed to the right): tail on the right; mode < median < mean.

• Illustrative examples from the text

  • Example 1 (left-skew): data 4,5,6,6,6,7,7,7,7,8; mean = 6.3, median = 6.5, mode = 7. Interpretation: mean is pulled toward the tail more than the median or mode.

  • Example 2 (right-skew): data 6,7,7,7,7,8,8,8,9,10; mean = 7.7, median = 7.5, mode = 7. Interpretation: the mean is the largest of the three, reflecting the right tail.

• Practical takeaway

  • When a distribution is skewed, the mean is more sensitive to outliers and the tail than the median or mode.

2.7 MEASURES OF THE SPREAD OF THE DATA

• Spread (variability) is a key characteristic of a data set.

• Standard deviation (the most common measure of spread)

  • Purpose: quantify how far data values are from their mean.

  • Interpretation: smaller SD indicates data are tightly clustered around the mean; larger SD indicates more dispersion.

  • The standard deviation is always nonnegative (positive or zero).

  • The SD is central to many statistical methods and helps assess whether a particular value is close to or far from the mean.

• Formulas (conceptual; see below for explicit forms):

  • For a sample: $$s = \