Statistics
Statistics is the branch of mathematics that collects numbers about the real world and then describes them in a few short figures. When a newspaper says that the average rainfall of Assam was so many centimetres, or that the most common mark in a test was fifty, it is using statistics. The examination asks three measures above all: the mean, the median and the mode, which together are called the measures of central tendency. They answer the same question in three different ways, telling us the typical value of a set of data. The questions are short and the arithmetic is elementary, so this chapter is one of the most reliable places to gain marks quickly; but it also demands neatness, because a single addition slip spoils the whole answer.
Basic Terms
- Data: a collection of numbers or observations gathered for a purpose.
- Frequency: the number of times a particular value appears in the data.
- Mean: the sum of all the observations divided by the number of observations.
- Median: the middle value when the observations are arranged in order.
- Mode: the value that appears most often.
- Range: the difference between the largest and the smallest observation.
- Cumulative frequency: the running total of the frequencies up to a given class.
The mean uses every observation and is therefore affected by extreme values, while the median and the mode are not. This difference explains why the median is preferred for incomes and house prices, where a few very large values would distort the mean.
The Core Formulas
- Mean of raw data = sum of the observations divided by their number.
- Mean with frequencies = sum of the products of value and frequency, divided by the sum of the frequencies.
- Mean by the assumed mean method = assumed mean plus the sum of the deviations divided by the number of observations.
- Median of an odd number of ordered observations = the value in the middle position.
- Median of an even number of ordered observations = the average of the two middle values.
- Median of grouped data = the lower limit of the median class plus the class width times the quantity, half the total frequency minus the cumulative frequency before the median class, divided by the frequency of the median class.
- Empirical relation: mode = three times the median minus two times the mean.
The empirical relation between the three measures is a convenient shortcut and is very often asked directly. Since it is only approximate, the examination usually arranges the numbers so that the relation holds exactly.
Finding the Mean
The mean is the most widely used measure because it uses every observation. For raw data it is simply the total divided by the count. When a frequency table is given, each value is multiplied by its frequency, the products are added, and the sum is divided by the total frequency. When the numbers are large, the assumed mean method is faster: a convenient value near the middle is taken as the assumed mean, the deviations of each observation from it are found, and the mean is the assumed mean plus the total deviation divided by the count. This method keeps the arithmetic small and reduces the chance of error.
Finding the Median
The median is the middle value, and therefore the data must first be arranged in ascending order. With an odd number of observations there is one middle value; with an even number the two middle values are averaged. The great advantage of the median is that it is not disturbed by extreme values. For grouped data the median lies in the class that contains the halfway point of the total frequency, and its exact position inside that class is found by interpolation, using the formula given above. A cumulative frequency table makes this quick, because the median class is the one where the running total first crosses half the total frequency.
Finding the Mode
The mode is the value or the class that occurs most often. For raw data it is found by inspection once the values are counted. For grouped data the modal class is the class with the highest frequency, and the exact value is obtained by a short interpolation involving the frequencies of the classes on either side. The mode is the most useful measure for categorical data such as shoe sizes or blood groups, where the idea of a middle value has little meaning. When two values occur equally often, the data is bimodal.
Range and Spread
The range is the simplest measure of how spread out the data is, being the difference between the highest and the lowest value. It is easy to compute but depends on only two observations, so a single unusual value can make it misleading. Questions that ask which set of data is more consistent are really asking which set has the smaller spread. The mean, the median and the mode describe the centre, while the range and similar measures describe the spread, and a complete description of data needs both.
Graphs and Their Meaning
Statistics are often presented as bar graphs, histograms, pie charts or line graphs. A bar graph compares different categories, a histogram shows how a continuous quantity is distributed across classes, a pie chart shows the share of each part in a whole, and a line graph shows how a quantity changes over time. In every case the candidate must read the scale carefully before calculating. A frequent mistake is to read the height of a bar as the value itself when the axis does not start at zero, which is why the scale of both axes should always be checked first.
How This Chapter Appears in the Examination
ADRE, APSC, Assam Police, SSC and Railway papers ask the mean, median and mode of small sets of data, the median and mode from a frequency table, and the use of the empirical relation. The questions are short but demand accurate addition, and the examiner sometimes hides the data inside a small table or a graph. Because the method is fixed, regular practice makes this one of the fastest chapters to complete.
Common Mistakes to Avoid
- Forgetting to arrange the data in order before finding the median.
- Dividing by the number of distinct values instead of the total frequency.
- Using the empirical relation with the terms in the wrong order.
- Reading a graph without first checking the scale of the axes.
- Mistaking the class with the largest value for the class with the highest frequency when finding the mode.
How to Prepare This Topic
- Learn the mean, median and mode of raw data and practise each on its own.
- Move on to frequency tables and the assumed mean method.
- Learn the median and mode of grouped data with a cumulative frequency table.
- Memorise the empirical relation and use it to check answers.
- Finish by reading data from tables, bar graphs and pie charts.
Frequently Asked Questions
What is the difference between the mean, median and mode?
The mean is the total divided by the number of observations, the median is the middle value after arranging the data, and the mode is the value that occurs most often. The mean uses every observation, while the median and mode are less affected by extreme values.
How do you find the median of an even number of observations?
Arrange the data in order and take the average of the two middle values. For example, the median of four, six, eight and ten is the average of six and eight, which is seven. For an odd number of observations the median is simply the single middle value.
What is the empirical relation between the three measures?
Mode equals three times the median minus two times the mean. It is an approximate relation, but examination questions are usually framed so that it holds exactly, and it can be used either to find a missing measure or to check an answer.
Which measure should be used for incomes and house prices?
The median is preferred, because a few very high values would pull the mean upward and give a false picture of the typical value. The median is not disturbed by such extreme observations, which is why it is used for wages, property prices and similar data.
How should I prepare statistics for ADRE and APSC?
Practise the mean, median and mode of raw data first, then move to frequency tables and grouped data, and finally learn to read tables and graphs. Always arrange the data before finding the median, and check every answer with the empirical relation whenever the three measures are involved.