Frequency distribution is an important tool in statistics which helps in organizing and summarizing large amounts of data. It arranges observations in a systematic manner and presents them in a tabular form. The correct option for the given question is D, i.e., all these. Let us understand each option in detail.

a) Arranges observations in an increasing order
This means that the observations are arranged from the smallest to the largest value. This is done to get a clear understanding of the data and to identify any patterns or trends. For example, if we have a data set of exam scores of 10 students, we can arrange them in an increasing order to see the range of scores and identify any outliers.

b) Arranges observation in terms of a number of groups
This means that the observations are grouped into categories or intervals. This is done when the data is too large or continuous to be presented in a simple tabular form. For example, if we have a data set of ages of a population, we can group them into intervals of 0-10, 11-20, 21-30, and so on. This helps in getting a better understanding of the age distribution of the population.

c) Relates to a measurable characteristic
This means that the data being presented is related to a measurable characteristic such as height, weight, age, income, etc. This characteristic can be measured quantitatively or qualitatively. For example, if we have a data set of heights of 100 individuals, we can present it in a frequency distribution table to see how many individuals fall into each height interval.

Therefore, a frequency distribution arranges observations in an increasing order, arranges observation in terms of a number of groups, and relates to a measurable characteristic.