Descriptive Statistics — Summarising Data in Numbers

Why We Summarise

The FoodHub dataset has 1,898 rows and 9 columns. That is 17,082 individual data points. No human can hold 17,082 numbers in their head, compare them, and draw conclusions. We need summaries — statistics that compress the data into a handful of numbers that capture its essential character.

But every summary loses information. When you report that the average order cost is $16.50, you have collapsed 1,898 different cost values into a single number. That number is useful, but it hides the fact that some orders cost $4.47 and others cost $35.47. The art of descriptive statistics is choosing which information to lose, and being honest about what the summary cannot tell you.

The two fundamental questions any summary must answer are:

Where is the centre? What is the typical value? What would you expect if you drew a random observation? This is answered by measures of central tendency.

How spread out is the data? Is every value close to the centre, or do values vary widely? This is answered by measures of spread.

A summary that reports only the centre without the spread is almost always misleading.

Measures of Central Tendency

There are three common measures of the "typical" value in a dataset. They agree when data is symmetric and disagree when it is skewed — and the disagreement is informative.

The Mean (Arithmetic Average)

The mean is the sum of all values divided by the count of values. It is the "balance point" of the distribution — the point at which the distribution would balance if you placed it on a fulcrum.

The mean uses every data point, which makes it sensitive to extreme values. A single order costing $200 in a dataset of $15 orders will pull the mean upward significantly. This sensitivity is both a strength (it reflects every observation) and a weakness (it can be distorted by extremes).

The Median

The median is the middle value when all observations are sorted in order. If there are an even number of observations, it is the average of the two middle values. Half of all observations fall below the median; half fall above.

Unlike the mean, the median is resistant to outliers. A single extreme value does not move the median much, because the median only cares about rank order, not magnitude. This makes the median a better measure of the "typical" value when distributions are skewed.

The Mode

The mode is the most frequently occurring value. It is most useful for categorical data ("the most common cuisine type is American") and discrete numeric data. For continuous data like order cost, the mode is often not meaningful — every value may be unique.

When Mean ≠ Median: Skewness is Present

In a perfectly symmetric distribution, mean = median = mode. When the mean and median diverge, the distribution is skewed. The direction of skew tells you about the tail:

In FoodHub, order cost has a mean of $16.50 but a median of $14.14. The mean is significantly higher — because a small number of expensive orders are pulling it up. If you told a restaurant partner "typical orders cost $16.50," you would be overestimating what most customers spend. The median ($14.14) is more honest.

Measures of Spread

Knowing the centre of a distribution is only half the picture. Two datasets can have identical means but completely different shapes — one tightly clustered around the mean, the other spread across a wide range. Measures of spread capture this variation.

Standard Deviation

The standard deviation (σ for a population, s for a sample) measures the average distance of each observation from the mean. A low standard deviation means most values are close to the mean; a high standard deviation means values are spread widely.

The formula squares the deviations from the mean (so positive and negative deviations don't cancel), averages them (this is the variance), and then takes the square root to return to the original units. The n − 1 in the denominator (rather than n) is Bessel's correction for sample data — it makes the estimate unbiased.

Variance

Variance is simply the standard deviation squared. It measures the same thing — spread around the mean — but in squared units, which makes it harder to interpret directly. However, variance is mathematically convenient (it can be added across independent variables), so it appears frequently in statistical theory even though we usually report standard deviation in practice.

Range

The range is simply the maximum value minus the minimum value: the full extent of the data. It is easy to compute and interpret, but it is highly sensitive to outliers — a single extreme value dramatically widens the range. For FoodHub order cost, the range is $35.47 − $4.47 = $31.00, which tells us something but doesn't capture where most values fall.

Interquartile Range (IQR)

The IQR is the range of the middle 50% of the data: the distance between the 25th percentile (Q1) and the 75th percentile (Q3).

The IQR is resistant to outliers — because it only looks at the middle half of the data, extreme values at either end do not affect it. For skewed distributions, reporting median ± IQR is more informative than reporting mean ± standard deviation, because neither the median nor the IQR is distorted by the skew.

Percentiles & Quartiles

A percentile tells you the value below which a given percentage of observations fall. The 25th percentile (Q1) is the value below which 25% of observations fall. The 50th percentile is the median. The 75th percentile (Q3) is the value below which 75% of observations fall.

Quartiles divide the data into four equal groups: the bottom 25%, the lower-middle 25%, the upper-middle 25%, and the top 25%. The box plot is a visual representation of this five-number summary:

The box plot gives you the full IQR story at a glance: where the bulk of the data sits, how symmetric the distribution is, and whether there are extreme values. Box plots are especially powerful for comparing distributions across groups — plotting delivery time by day of week, for instance, puts six box plots side by side and makes differences immediately visible.

Why Q1/Q3/IQR Beat Min/Max

The minimum and maximum values are the most extreme observations in the dataset. They are highly sensitive to outliers and data errors, and they tell you nothing about where most of the data falls. Q1 and Q3 are more stable, more representative, and more useful for understanding the typical spread of data.

Consider FoodHub delivery time: min = 5 min, max = 83 min, range = 78 min. That range is dramatic. But Q1 = 15 min, Q3 = 31 min, IQR = 16 min. The middle half of all deliveries falls in a 16-minute window. The range of 78 minutes is driven by a small number of unusual orders — the IQR tells the real story.

Skewness

Skewness describes the asymmetry of a distribution. A symmetric distribution (like the normal distribution) has zero skewness — the left and right tails are mirror images. A skewed distribution has one tail longer than the other.

In FoodHub, order cost is right-skewed: most orders are relatively affordable, but a small number of expensive orders pull the mean ($16.50) above the median ($14.14). Ratings are left-skewed: most customers who rate give 4 or 5 stars, pulling the median (4.5) above the mean (4.34). For right-skewed data, always report the median rather than the mean as the "typical" value.

The Danger of the Average

There is a classic joke in statistics: "A statistician drowned crossing a river with an average depth of 3 feet." The joke is a serious warning about the limits of means.

If a river is 1 foot deep for 90% of its width and 30 feet deep in one channel, the average depth is 3.9 feet. That average is mathematically correct and completely useless for deciding whether to wade across. The distribution — specifically the existence of the 30-foot channel — is the information that matters, and the average hides it.

The same logic applies to business data. Consider these scenarios where the mean misleads:

• 1Bimodal distribution: A food delivery app serves both quick lunch orders (~15 min) and elaborate dinner orders (~45 min). The mean delivery time of 30 minutes describes neither group. There is no order that takes "average" time — the average is in the gap between the two modes.
• 2Extreme outliers: If one restaurant received 1,000 orders and all others received 10–50, the mean orders-per-restaurant is inflated to something no typical restaurant achieves.
• 3Aggregation across groups: The overall mean rating might be 4.3, but if one cuisine type averages 3.8 and another averages 4.7, the overall mean obscures the important difference.