Descriptive statistics

Descriptive statistics are numbers that summarize and describe the data you have, without inferring anything beyond it (Lane). Four groups of numbers are in common use: the mean, the median, the standard deviation with the variance, and quantiles.

What do the numbers of descriptive statistics tell you, and what are they for?

Take the hourly returns of BTCUSDT (bitcoin traded against USDT on Binance) from January 1, 2025 to August 14, 2026 as an example. An hourly return is the percentage change of the closing price from the previous hour, computed on 1-hour candles in UTC, which gives 14,184 values. The histogram below sorts the values into intervals 0.05% wide and counts the hours in each. It marks the mean, the median, the middle half of the data (Q1 to Q3) and the range of the mean ± 1 standard deviation.

Chart
Python code

import numpy as np
import pandas as pd
import requests


def utc_ms(time):
    return int(pd.Timestamp(time, tz="UTC").timestamp() * 1000)


def hourly_closes(symbol, start, end):
    """Closing prices of the 1-hour candles opened from start to end (UTC), at most 1,000 candles per request."""
    closes, t = [], utc_ms(start)
    while t <= utc_ms(end):
        klines = requests.get(
            "https://data-api.binance.vision/api/v3/klines",
            params={"symbol": symbol, "interval": "1h", "startTime": t, "endTime": utc_ms(end), "limit": 1000},
        ).json()
        closes += [float(k[4]) for k in klines]
        t = klines[-1][0] + 3_600_000   # open of the next candle
    return pd.Series(closes)


close = hourly_closes("BTCUSDT", "2024-12-31 23:00", "2026-08-14 23:00")   # 14,185 closing prices
ret = close.pct_change().dropna() * 100                                     # hourly returns (%): 14,184 values

x = np.sort(ret.to_numpy())                                  # x(1) <= x(2) <= ... <= x(N)
N = len(x)
mean = x.sum() / N
median = x[N // 2] if N % 2 else (x[N // 2 - 1] + x[N // 2]) / 2
var = ((x - mean) ** 2).sum() / (N - 1)
std = var ** 0.5


def quantile(p):
    h = p * (N - 1)                                          # position in x, counting from 0
    j = int(h)
    return x[j] + (h - j) * (x[j + 1] - x[j])                # interpolate between two adjacent values


q1, q3 = quantile(0.25), quantile(0.75)
print(N, round(mean, 4), round(median, 4))                   # count, mean, median: 14184 -0.0017 0.0008
print(round(var, 4), round(std, 4))                          # variance and standard deviation: 0.2233 0.4725
print(round(q1, 4), round(q3, 4), round(q3 - q1, 4))         # Q1, Q3 and IQR: -0.1983 0.2022 0.4004
print(round(ret.mean(), 4), round(ret.median(), 4), round(ret.var(), 4), round(ret.std(), 4), ret.quantile([0.25, 0.75]).round(4).tolist())   # pandas gives the same numbers: -0.0017 0.0008 0.2233 0.4725 [-0.1983, 0.2022]
print(round(ret.std(ddof=0), 4))                             # standard deviation dividing by N instead of N - 1: 0.4725
print(round(ret.min(), 2), round(ret.max(), 2))              # smallest and largest: -4.9 5.03

extreme = ret.abs() > 2                                      # hours in which the price moved by more than 2%
kept = ret[~extreme]
print(int(extreme.sum()))                                    # number of hours with a move above 2%: 96
print(round(((ret[extreme] - ret.mean()) ** 2).sum() / ((ret - ret.mean()) ** 2).sum(), 2))   # their share of the sum of squared deviations: 0.22
print(round(kept.mean(), 4), round(kept.median(), 4), round(kept.std(), 4), round(kept.quantile(0.75) - kept.quantile(0.25), 4))   # without them: mean, median, standard deviation, IQR: -0.0004 0.001 0.4183 0.3974

counts, _ = np.histogram(ret, bins=np.linspace(-2, 2, 81))   # hours in 80 intervals 0.05% wide from -2% to 2%
print(counts.tolist(), int((ret < -2).sum()), int((ret > 2).sum()))   # hours in each interval, then hours below -2% and above 2%: [5, 4, 5, ..., 3] 51 45

Let

x_1, x_2, \dots, x_N

be the

N

values, here

N = 14\,184

returns,

\sum

the sum over all

N

values, and

x_{(1)} \le x_{(2)} \le \dots \le x_{(N)}

the same values sorted from smallest to largest. For quantiles,

p

is a proportion between 0 and 1, and

j

and

g

are the integer part and the fractional part of

p(N - 1) + 1

, so

Q_p

interpolates between two adjacent values. The table gives the formula and meaning of each number.

Number	Formula	Meaning
Mean	$\bar{x} = (x_1 + x_2 + \dots + x_N) / N$	The center of the data. Every value contributes, so extreme values pull it
Median	$N$ odd: $x_{((N+1)/2)}$ $N$ even: $(x_{(N/2)} + x_{(N/2+1)}) / 2$	The middle value: half the data lies below it. It depends only on order, so an extreme value, however large, takes up just one position and does not shift it
Variance	$s^2 = \sum (x_i - \bar{x})^2 / (N - 1)$	The spread of the data around the mean. It is computed from squared deviations, so its unit is %², not the unit of the data
Standard deviation	$s = \sqrt{s^2}$	The square root of the variance, which brings the spread back to the unit of the data (%). Extreme values inflate it
Quantile (percentile, quartile)	$Q_p = x_{(j)} + g\,(x_{(j+1)} - x_{(j)})$	The proportion $p$ of the data lies below $Q_p$ . Q1 is $Q_{0.25}$ and Q3 is $Q_{0.75}$
IQR	$\text{IQR} = Q_3 - Q_1$	The spread of the middle half of the data, so extreme values barely change it

On the 14,184 returns, the mean is −0.0017%, the median +0.0008%, the standard deviation 0.4725% (variance 0.2233 %²), Q1 −0.1983%, Q3 +0.2022% and the IQR 0.4004 percentage points. Try dropping the 96 hours in which the price moved by more than 2%, the hours outside the frame of the chart. They are only 0.7% of the hours but make up 22% of the sum of squared deviations, so the standard deviation falls by more than 11% (to 0.4183%) while the IQR falls by less than 1% (to 0.3974%). The mean changes to −0.0004%, and the median edges up to +0.0010%. Divide the variance by

N - 1

when the data is a sample from a larger set, since dividing by

N

tends to give a result lower than the variance of the whole set (Lane); divide by

N

when the data is the whole set. NumPy divides by

N

by default and pandas by

N - 1

, but with 14,184 values both give 0.4725%.

References

Sources: Lane, Online Statistics Education (Rice University), the NIST/SEMATECH e-Handbook sections 1.3.5.1 and 1.3.5.6, and the documentation of NumPy and pandas. Prices come from Binance’s public API.

Law of large numbers

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Foundations

Descriptive statistics

What do the numbers of descriptive statistics tell you, and what are they for?

References

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Law of large numbers

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Law of large numbers

Expected value

What do the numbers of descriptive statistics tell you, and what are they for?

References

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