Describing and Displaying Data

What are the three main ways to summarize empirical distributions?

Tables (frequency distributions), graphs (bar graphs, histograms), and descriptive statistics.

What does "location" refer to in a distribution?

Where the distribution lies on a number line (e.g., x-axis of a histogram).

What is central tendency?

It refers to the "middle" of a distribution.

What is the mode?

The most frequently occurring value(s) in a distribution.

Is the mode affected by outliers?

No, it is robust to outliers.

What do unimodal, bimodal, and multimodal mean?

They describe the number of modes (1, 2, 3+, respectively).

What is the median?

The 50th percentile — the value that splits the distribution in half by frequency.

How do you calculate the median?

Rank the values and use the middle score, or average the two middle scores if N is even.

Is the median affected by outliers?

No, it is relatively robust to outliers.

What is the median locator formula?

(N + 1) ÷ 2

What is the mean?

The arithmetic average of all values.

What does the mean minimize?

The sum of squared deviations.

Is the mean sensitive to outliers?

Yes, outliers can significantly affect the mean.

What does dispersion measure?

How spread out the values are from the center (mean).

What is the range formula?

Max value − Min value

What is the IQR formula?

IQR = 75th percentile − 25th percentile

What does IQR represent?

The spread of the middle 50% of data.

What is variance?

The average of the squared deviations from the mean.

What’s the formula for sample variance (data analysis)?

What’s the formula for sample variance (inference)?

s2=∑(Xi​−Xˉ)2​
——————-

N

Why do we square the deviations?

To prevent them from canceling out (∑ deviations = 0).

What is standard deviation?

The square root of the variance.

Why is standard deviation preferred over variance?

It’s in the same metric as the original data.

What does skewness describe?

The symmetry of a distribution.

What does g1=0g1​=0 indicate?

Symmetric distribution.

What does g1>0g1​>0 mean?

Positive skew (tail to the right).

What does g1<0g1​<0 mean?

Negative skew (tail to the left).

What does kurtosis describe?

The peakedness of a distribution.

What does g2=0g2​=0 indicate?

Mesokurtic (normal distribution shape).

What does g2>0g2​>0 indicate?

Leptokurtic (tall/narrow peak).

What does g2<0g2​<0 indicate?

Platykurtic (flat/wide peak).

What do boxplots visualize?

Dispersion and outliers in a distribution.

What is the max length of whiskers in a boxplot?

Up to 1.5 × IQR from Q1 or Q3.

What are aggregate-type propositions?

Claims about groups as a whole.

Why should we be cautious with individual-level interpretations in statistics?

Because statistics summarize groups, not individuals.