What is the difference between sample and population standard deviation?

Population standard deviation divides by N (all data points). Sample standard deviation divides by N−1 (Bessel's correction) to account for the fact that a sample underestimates the true spread of the full population.

The median is the middle value when all numbers are sorted in order. For an even count of numbers, it is the average of the two middle values.

What does "no mode" mean?

If every number appears exactly once, there is no mode. Mode is only reported when at least one value appears more than once.

The interquartile range (IQR) is Q3 − Q1. It measures the spread of the middle 50% of your data and is useful for identifying outliers.

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Standard deviation explained

Standard deviation measures how spread out the values in a dataset are from the mean. A low standard deviation means values cluster close to the mean; a high standard deviation means they are spread out widely.

Use sample standard deviation (÷ n−1) when your data is a sample drawn from a larger population. Use population standard deviation (÷ n) when you have data for the entire population.

Sample vs population statistics

Statistic	Sample formula	Population formula
Variance	Σ(x − x̄)² / (n−1)	Σ(x − μ)² / n
Std dev	√(sample variance)	√(population variance)

Quartiles and IQR

Quartiles divide your sorted data into four equal parts. Q1 is the 25th percentile, Q3 is the 75th percentile, and the interquartile range (IQR = Q3 − Q1) measures the spread of the middle 50% of the data. Values more than 1.5 × IQR below Q1 or above Q3 are often considered outliers.

Measures of central tendency

Mean: the sum of all values divided by the count. Best for symmetric distributions without outliers. Sensitive to extreme values - a single outlier can pull the mean far from the center of the data.
Median: the middle value when sorted. Robust to outliers and better represents "typical" in skewed distributions (e.g., income data, where a few very high earners distort the mean).
Mode: the most frequently occurring value. Most useful for categorical data or when identifying the most common outcome.

Skewness and kurtosis

Skewness measures the asymmetry of the distribution. A positive skew (right skew) has a long tail to the right and the mean is pulled above the median - income and wealth distributions are classic examples. A negative skew (left skew) has the tail to the left.

Kurtosis measures the "tailedness" of the distribution. High kurtosis (leptokurtic) indicates heavy tails and more extreme outliers than a normal distribution. Low kurtosis (platykurtic) indicates lighter tails. A normal distribution has a kurtosis of 3 (excess kurtosis of 0).

Z-score

A Z-score expresses how many standard deviations a value is from the mean:

z = (x − μ) / σ

A Z-score of +2 means the value is 2 standard deviations above the mean. In a normal distribution, about 95% of values fall within ±2 standard deviations (Z-score between −2 and +2). Z-scores enable comparison of values from different datasets with different units and scales.

Outlier detection

IQR method: values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR are considered mild outliers. Using 3×IQR as the threshold identifies extreme outliers. This method is robust because IQR itself is not affected by outliers.
Z-score method: values with |Z| > 3 are typically flagged as outliers (about 0.3% of values in a normal distribution). This method is sensitive to extreme outliers - they inflate the standard deviation, which can mask other outliers.