Descriptive Statistics

Suppose you are looking at last month's order data for an online store. The result shows that the average order amount for 1,000 customers is $100. Is this number alone enough to understand how much customers usually spend?

In reality, most customers may spend $40–$60 while a few customers pay several hundred dollars. In that case, the average order amount is $100, but what a typical customer spends may be lower than that.

You also need to check how much order amounts differ from customer to customer, whether orders are concentrated in a particular price range, and whether there are unusually large orders.

A method for summarizing the overall features of data and understanding its distribution like this before analyzing it in earnest is called descriptive statistics. Descriptive statistics organize the center, spread, and shape of the distribution of data with numbers, tables, and graphs.

In this article we look at the main descriptive statistics and how they are calculated, a worked analysis example, interpreting the results table, handling outliers and missing values, and descriptive statistics by group.

For example, instead of checking the order amounts of 1,000 customers one by one, you can quickly understand the main features of the data by computing the mean, median, standard deviation, minimum, maximum, and so on. Descriptive statistics look at data from three main perspectives.

The main purpose of descriptive statistics is to describe the observed data. So it is not a method for concluding that there is a significant difference in the population because there is a difference in means, or for claiming causation because two variables are related.

Questions like these may require separate inferential methods such as t-tests, analysis of variance, correlation analysis, and regression analysis.

The best-known are the mean, median, and mode. For example, suppose the order amounts of 7 customers are as follows.

Mean

The sum of all values divided by the number of observations.

The mean order amount is $100.

Median

The value in the middle when the data are sorted by size. In this example, the fourth value, $60, is the median.

Mode

The value that appears most often. $50 appears twice, so the mode is $50.

The three measures differ because the large order amount of $350 affected the mean. The mean has the advantage of reflecting every observation, but it is sensitive to extreme values. The median, on the other hand, is relatively less affected by extreme values, so it is useful for describing data whose distribution is skewed to one side.

For example, suppose the order amounts at two stores on one day are as follows.

The average order amount at both stores is $100. But at store A, order amounts cluster around $100, while at store B they are spread widely from $20 to $180. The measures that express this difference in numbers are the variance and the standard deviation.

Variance

A value that summarizes how far each observation is from the mean by squaring those distances. The sample variance is computed as follows.

Standard deviation

The square root of the variance.

The sample standard deviations for the example above are as follows.

Store B's standard deviation is much larger, which tells us its order amounts are more widely spread. Because the standard deviation uses the same units as the original data, it is often easier to interpret than the variance. However, it can be affected by extreme values, so for data with a heavily skewed distribution it is good to check the interquartile range as well.

Typically you check the following three values.

• First quartile (Q1): the point below which the lowest 25% of the data fall
• Second quartile (Q2): the median, the 50% point
• Third quartile (Q3): the point below which the lowest 75% of the data fall

The interquartile range (IQR) is Q3 minus Q1.

For example, suppose the distribution of order amounts for an online store's customers is as follows.

The IQR is $116 minus $44, which is $72. This is the width of the range that contains the middle 50% of the data. Quartiles are relatively less affected by extreme values, so they are useful for distributions with a long right tail, such as order amounts.

However, the way quartiles are computed can differ slightly between statistical programs, so if exact reproduction is needed, it is good to check which quantile method was used.

The main measures are skewness and kurtosis.

Skewness

Skewness indicates in which direction a distribution is skewed.

• If skewness is close to 0, the distribution is roughly symmetric.
• If skewness is positive, the distribution tends to have a long right tail.
• If skewness is negative, the distribution tends to have a long left tail.

Order amounts at an online store can have a right-skewed distribution, because a few customers sometimes pay very large amounts.

Kurtosis

Kurtosis is a measure related to the tail behavior of a distribution, especially how often extreme values appear. It is commonly computed as excess kurtosis relative to the normal distribution, and the excess kurtosis of a normal distribution is 0. However, it is not appropriate to interpret kurtosis simply as a measure of how sharp the peak of a distribution is.

Skewness and kurtosis can be affected by sample size and extreme values. So rather than judging normality from these two measures alone, it is good to also check a histogram, a Q-Q plot, and the like.

The researcher wants to know the overall features of order amounts and delivery times.

Analysis data

The research question is as follows.

For order amount and delivery time, you can compute the mean, median, standard deviation, minimum, maximum, and so on. The number of items per order is a numeric variable, so you can check its mean and distribution, but presenting the frequency of each item count as well makes it easier to understand. Membership tier and payment method are categorical variables, so it is common to compute frequencies and percentages.

Descriptive statistics for the main numeric variables

These values were not computed from real store data.

The mean order amount is $104.80 and the median is $76. Because the mean is higher than the median, relatively large order amounts may have pulled the mean up. However, the difference between the mean and the median alone cannot fully determine the shape of the distribution, so it is good to check a histogram or box plot as well.

The mean delivery time was 3.4 days and the median was 3 days. This shows that typical delivery takes about 3 days. Delivery time has 980 valid observations, meaning that delivery time information for 20 of the 1,000 orders is missing or unusable for analysis.

The number of items per order had a mean of 2.6 and a median of 2. So it can be summarized that customers bought about 2 to 3 items per order on average.

Histogram

A histogram shows the distribution of numeric data by interval. For example, you can divide order amounts into $20 or $40 intervals and show the number of orders in each interval. This lets you see whether orders are concentrated in a particular price range and whether the distribution has a long tail to the right or the left.

The last interval is an open interval with a different width, so this bar chart is not meant for precise histogram density comparisons.

Box plot

A box plot shows the median, quartiles, outlier candidates, and so on at a glance. It is good for checking whether there are some customers with high order amounts and over what range the middle 50% of the data are distributed. Also, when comparing order amounts by membership tier, placing box plots for each group side by side makes it easy to see differences in each group's median and distribution.

However, an outlier candidate shown on a graph does not mean the data point is actually an error.

For example, if most order amounts are $20–$200 but some orders exceed $800, you can flag them as outlier candidates. A typical detection method is the IQR rule. Values outside the following range are generally flagged as outlier candidates.

For example, suppose Q1 is $44 and Q3 is $116. The IQR is $72, so the upper threshold is as follows.

So observations with order amounts above $224 can be flagged as outlier candidates.

So when you find outliers, first check whether they are entry errors, special transactions such as refunds or duplicate payments, or normal transactions that actually happened. If needed, you can report both the mean and the median, or run a sensitivity analysis that includes and excludes the extreme values.

For example, there may be 1,000 orders in total but only 980 with a recorded delivery time. In this case, the mean and standard deviation of delivery time are generally computed from the 980 usable orders. So it is good to show not only the total N but also the valid N for each variable in the results table.

The missing rate is computed as follows.

The missing rate for delivery time is 20 ÷ 1,000 × 100 = 2%. However, a low missing rate does not necessarily mean the bias is small. For example, if delivery completion dates were more likely to be missing for orders that took a very long time to deliver, the actual mean delivery time could be underestimated.

So if an important variable has missing values, you should check why the missingness occurred and what pattern it follows.

For example, you can check how the mean and median order amount differ by membership tier.

Order amount by membership tier

Gold members had the highest mean order amount and standard members the lowest. These results show that the observed level of order amounts differs by membership tier. However, descriptive statistics alone cannot determine whether these differences are statistically significant in the population.

To test differences in mean order amount between membership tiers, you can consider analysis of variance or nonparametric tests, depending on the data structure and analysis assumptions. Also, a finding that higher-tier customers spend more should not lead to the conclusion that the membership tier itself increases order amounts.

Even with the same mean, the median, standard deviation, and shape of the distribution can differ greatly. So for important numeric variables it is good to check several summary measures together.

Don't delete extreme values unconditionally.

If you find very large or small values, first check whether they are actual data errors. Removing observations just because they are outlier candidates can distort the results.

Check for missing values.

The number of valid observations can differ by variable, so distinguish the total sample size from the sample size actually used in each calculation.

Check the units of measurement.

The interpretation of the mean and standard deviation changes depending on whether order amounts are recorded in dollars or in thousands of dollars. Also, a mean or standard deviation may not be appropriate for categorical variables.

Don't interpret descriptive results as causal.

Even if differences between groups are observed, such as differences in order amount by membership tier, you can't conclude that the difference was caused by a particular factor.

Distinguish sample statistics from population parameters.

The sample mean and sample standard deviation describe the characteristics of the sample collected. To generalize these results to the whole population, you must consider how the sample was drawn and how representative it is.

When the distribution is skewed or extreme values matter, you can also report the quartiles and the IQR. Using the online store example above, you could write it as follows.

In real research, you can also present histograms or box plots to show the shape of the distribution of important variables. And in studies comparing groups, it is good to summarize each group's sample size and mean and standard deviation, or median and IQR, in a table.

The mean and median show the level around which the data mostly lie, and the standard deviation and interquartile range show how spread out the observations are. Skewness, kurtosis, histograms, box plots, and so on also let you check whether the distribution is skewed to one side and whether there are extreme values.

Also, checking missing values, outliers, and the number of valid observations for each variable lets you identify features of the data that need care in the statistical analyses that follow. However, descriptive statistics are a way of summarizing observed data, so they do not directly determine the statistical significance of group differences or causal relationships between variables.

So it is good to understand the features of the data from descriptive statistics first and then choose an inferential method that fits the research question.