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Intervals

When dealing with many uneven observations, the data can be divided into groups, also known as intervals.

With grouped observations, it is typically not possible to determine the mode, maximum value, minimum value, or range because the individual observations are unknown, and only the data is summarized in a frequency table. However, one can discuss a modal interval, which is the interval with the highest frequency of observations. One can also find the mean, median, and quartiles, but this is usually done differently for grouped observations.

Often, intervals are denoted with square brackets, "[" and "]". These brackets indicate whether the number is included. If the bracket faces the number, the number is included. If the bracket faces away from the number, it means the number is not included, but all numbers below it are included.

• For example, in the interval [2;4[, the number 2 is included, and all numbers up to 4 are included, but the number 4 itself is not included. This means that 3.999999999999999999999999, etc., is included. Thus, one can say "from 2 and up to, but not including 4."

Mean Relative to the Interval Midpoint

To find the mean of observations grouped in intervals, where you cannot return to the original observations, you first find the interval midpoint. This is the middle value of the interval. For instance, if the interval ranges from 0 to 10, the midpoint is 5. The midpoint is determined because the distribution of observations within the interval is unknown, so it is assumed that observations are evenly distributed around the midpoint.

If the individual observations were known, one would sum them and then divide by the total number. In fact, a similar approach is taken with observations in intervals. However, it is easier to multiply the interval midpoints.

• For example, if the interval midpoint is 5 and the frequency of the interval is 3, this corresponds to having the observations 5, 5, and 5. Therefore, it is easier to say 5 times 3 than 5 + 5 + 5.
• The numbers obtained for each interval are summed and divided by the total number of observations (not the number of intervals).

Observation diagram for grouped observations: