Describing and quantifying data Notes

Why We Describe And Quantify Data

1. When you collect data from a class survey, a science experiment, or a sustainability study (for example, energy use in different homes), the raw list of values is rarely useful on its own.
2. Describing and quantifying data means turning those values into clear summaries and representations so you can spot relationships, trends, and unusual results.
3. Two big ideas run through this topic:
   1. Where the data tends to be (typical values), and
   2. How spread out the data is (variation).

Data Type Determines What You Can Calculate And Draw

Before calculating any statistics, identify the type of data, because this affects which summaries and graphs are appropriate.

Quantitative data can be split into:

Organizing Raw Data So Patterns Become Visible

Frequency Tables And Grouped Frequency Tables

1. A frequency table records how often each value (or category) occurs.
2. For discrete quantitative or qualitative data, you can usually list each value/category.
3. For continuous data, or when there are many possible values, you often use grouped data, where values are placed into class intervals (for example, 150 to <160 cm).

Stem-And-Leaf Diagrams Preserve Individual Values

1. A stem-and-leaf diagram is a neat way to organize numerical data while still showing every value.
   1. The stem represents the leading digit(s).
   2. The leaf represents the final digit(s).
   3. A key explains how to read the numbers.
2. This makes it easy to see the shape of the distribution and quickly identify the median and quartiles.

Measures Of Central Tendency Describe A Typical Value

The three most common measures are mean, median, and mode.

Mean

1. The mean is the arithmetic average.
2. For ungrouped data with $n$ values $x1, x2, \dots, x_n$: $$\bar{x}=\frac{x1+x2+\cdots+x_n}{n}$$
3. For a frequency table (ungrouped): $$\bar{x}=\frac{\sum (x\times f)}{\sum f}$$
4. For a grouped frequency table, the exact mean cannot be found because the individual values are unknown.
5. Instead, you estimate it using the class midpoint $m$ for each interval: $$\bar{x}\approx\frac{\sum (m\times f)}{\sum f}$$

Median

1. The median is the middle value when the data is ordered.
2. If $n$ is odd, the median is the $\frac{n+1}{2}$th value.
3. If $n$ is even, the median is the mean of the $\frac{n}{2}$th and $\left(\frac{n}{2}+1\right)$th values.
4. The median is also called the second quartile, $\mathbf{Q_2}$.

Mode

1. The mode is the most frequent value.
2. A data set can have one mode, more than one mode, or no mode.
3. For grouped data, you may refer to the modal class (the class interval with the highest frequency).

Measures Of Dispersion Describe Variation In The Data

Range

1. The range measures the total spread: $$\text{range}=\text{max}-\text{min}$$
2. It is quick to calculate but is heavily affected by extreme values.

Quartiles And The Interquartile Range

1. Quartiles split ordered data into four equal parts.
   1. The lower quartile $\mathbf{Q_1}$ is the median of the observations to the left of the median.
   2. The upper quartile $\mathbf{Q_3}$ is the median of the observations to the right of the median.
2. The interquartile range (IQR) measures the spread of the middle 50% of data: $$\text{IQR}=Q_3-Q_1$$

Outliers Using The 1.5×IQR Rule

1. An outlier is a data value that does not fit the general pattern of the rest.
2. A point is an outlier if it is:
   1. less than $Q_1-1.5\times\text{IQR}$, or
   2. greater than $Q_3+1.5\times\text{IQR}$.

Box-And-Whisker Plots Summarize And Compare Distributions

A box-and-whisker plot (or box plot) is a diagram based on the five-point summary.

1. The box runs from $Q_1$ to $Q_3$.
2. The line inside the box is the median ($Q_2$).
3. The whiskers extend to the minimum and maximum (or sometimes to the last non-outlier values, depending on convention).

Grouped Data And Cumulative Frequency Curves Let You Estimate Quartiles And Percentiles

When data is grouped, you often use a cumulative frequency curve (also called an ogive) to estimate the median, quartiles, and other percentiles.

Building A Cumulative Frequency Table (Idea)

1. From a grouped frequency table, you add frequencies as you go to create cumulative totals.
2. You then plot:
   1. the upper class boundary on the horizontal axis, and
   2. the cumulative frequency on the vertical axis.
3. Join the points with a smooth increasing curve.

Reading The Five-Point Summary From A Cumulative Frequency Curve

1. Suppose there are $N$ total observations.
   1. Minimum: estimated from where the curve starts (near cumulative frequency 0).
   2. Maximum: estimated from where the curve ends (near cumulative frequency $N$).
   3. Lower quartile $Q_1$: the value at cumulative frequency $0.25N$.
   4. Median $Q_2$: the value at cumulative frequency $0.50N$.
   5. Upper quartile $Q_3$: the value at cumulative frequency $0.75N$.
2. To find each quartile, draw a horizontal line from the chosen cumulative frequency level to the curve, then drop a vertical line to the value axis.

Percentiles

1. A percentile is a value below which a given percentage of data falls.
2. The $k$th percentile corresponds to cumulative frequency $\frac{k}{100}N$.
3. For example, the 90th percentile is read at $0.90N$.

How Real Data Can Be Misleading

Even when calculations are correct, conclusions can still be unreliable if the data collection or representation is poor.

Common issues include:

1. Biased sampling (surveying only one group)
2. Inappropriate grouping (class widths that hide variation)
3. Cherry-picking measures (reporting the mean when the median would be more representative)
4. Ignoring outliers without justification