Statistics — Study Notes

1. Introduction

Statistics is the science of collecting, organising, analysing, and interpreting numerical data. In this chapter, students learn to summarise large sets of data using measures of central tendency: the mean, median, and mode. These three measures each describe the "centre" of a data set in a different way, and together they give a complete picture of a typical value in the data.

The chapter deals with grouped data, where raw observations are arranged into class intervals with frequencies. For grouped data, the mean is computed by the direct method, assumed-mean method, or step-deviation method; the mode is found using the modal class and an interpolation formula; and the median is found using the cumulative frequency distribution and a similar formula. The empirical relationship between the three measures is also presented.

Statistics appears everywhere in modern life, from election results and weather reports to school results and economic indicators. The ability to interpret and compute mean, median, and mode is a fundamental skill, and this chapter is a reliable source of marks in the board examination, typically featuring one long-answer question requiring all three measures or a combination of them.

2. Measures of Central Tendency for Ungrouped Data

For ungrouped data x1, x2, ..., xn:

Mean

mean = (x1 + x2 + ... + xn)/n

Median

Arrange the data in ascending order. If n is odd, the median is the middle term, the ((n + 1)/2)th term. If n is even, it is the average of the two middle terms, the (n/2)th and (n/2 + 1)th terms.

Mode

The mode is the value that occurs most frequently. A data set may have one mode, more than one mode, or no mode at all if all values occur equally often.

Example: For the data 2, 5, 3, 5, 7, 8, 5, Mean = (2 + 5 + 3 + 5 + 7 + 8 + 5)/7 = 35/7 = 5. Median: sorted data 2, 3, 5, 5, 5, 7, 8; middle term is 5. Mode = 5 (occurs three times).

3. Mean of Grouped Data

For grouped data with class marks xi and frequencies fi, the mean is: Mean = (sum of fi xi)/(sum of fi)

Methods

1. Direct method: compute fi xi for each class and sum.
2. Assumed-mean method: choose an assumed mean a, define di = xi - a, then mean = a + (sum fi di)/(sum fi).
3. Step-deviation method: define ui = (xi - a)/h where h is the class size, then mean = a + h x (sum fi ui)/(sum fi).

Example: Classes 0-10, 10-20 with marks 5, 15 and frequencies 3, 7: Direct method: sum fi xi = 3 x 5 + 7 x 15 = 15 + 105 = 120, sum fi = 10, mean = 12.

The step-deviation method is the fastest when class marks are large, because it reduces them by dividing by the class size.

4. Mode of Grouped Data

The mode of grouped data is computed using the modal class, which is the class interval with the highest frequency. If the lower limit of the modal class is l, its frequency is f1, the frequency of the preceding class is f0, the frequency of the following class is f2, and the class size is h, then: Mode = l + [(f1 - f0)/(2f1 - f0 - f2)] x h

Example: For the modal class 60-70 (l = 60, h = 10) with f1 = 20, f0 = 12, f2 = 8: Mode = 60 + [(20 - 12)/(40 - 12 - 8)] x 10 = 60 + (8/20) x 10 = 60 + 4 = 64.

The mode formula interpolates within the modal class to locate the most frequent value.

5. Median of Grouped Data

For grouped data, we use cumulative frequencies. If n/2 is less than the cumulative frequency of a class, that class is the median class. Let l be the lower limit of the median class, f its frequency, cf the cumulative frequency of the class preceding it, and h the class size. Then: Median = l + [(n/2 - cf)/f] x h

Cumulative Frequency

The cumulative frequency of a class is the sum of the frequencies of all classes up to and including that class.

Example: If n/2 = 25 and the median class 30-40 has l = 30, cf = 12, f = 16, h = 10: Median = 30 + [(25 - 12)/16] x 10 = 30 + (13/16) x 10 = 30 + 8.125 = 38.125.

Empirical Relationship

For a moderately skewed distribution: Mode = 3 x Median - 2 x Mean

This relationship is used to estimate any one of the three measures when the other two are known.

6. Cumulative Frequency Curves (Ogive)

An ogive is the graph of the cumulative frequency distribution. Two types exist: - Less than ogive: plotted using upper limits against cumulative frequencies. - More than ogive: plotted using lower limits against cumulative (greater than) frequencies.

The point where the less-than and more-than ogives intersect gives the median. Alternatively, a horizontal line at n/2 on the less-than ogive meets the curve at a point whose x-coordinate is the median.

Drawing an ogive accurately is a standard board question that tests graphical skills and interpretation.

Quick Revision Tables

Table 1: Measures of Central Tendency for Grouped Data

Table 2: Relationship Among Measures

Mind Map

Important Diagrams (SVG)

Diagram 1: Ogive Showing Median

Diagram 2: Mean, Median and Mode on a Frequency Curve

Common Mistakes

1. Forgetting to sort the data before finding the median of ungrouped data; the median requires the ascending order.
2. Using (n + 1)/2 for an even number of observations, which lands between two positions instead of averaging the two middle terms.
3. Choosing the wrong class as the modal or median class because the highest frequency is read incorrectly.
4. Confusing f1, f0, and f2 in the mode formula: f1 is the modal class frequency, f0 the previous, f2 the next.
5. Forgetting to multiply by the class size h in the step-deviation mean, or dividing by the wrong class size.
6. In the median formula, using the cumulative frequency of the median class itself instead of the cumulative frequency cf of the preceding class.
7. Mixing up less-than and more-than ogives when plotting, or reading the median from the wrong curve.

Exam Tips

1. For grouped mean, set up columns for xi (class mark), fi, and fi xi neatly; method marks are awarded for the table.
2. Choose the assumed mean from the middle of the table to simplify arithmetic in the assumed-mean method.
3. The step-deviation method is fastest when class marks are large; remember to multiply the final correction by h.
4. Identify the modal and median classes first, then substitute into the formula with careful attention to l, h, f1, f0, f2, and cf.
5. Use the empirical relation Mode = 3 Median - 2 Mean to cross-check computed values.
6. When drawing an ogive, plot cumulative frequencies against correct boundaries and use a smooth curve.
7. Practise completing missing frequencies in a table, as these two-mark questions are common and formula-heavy.

Conclusion

Statistics equips students with the tools to summarise and understand data using the mean, median, and mode. For grouped data, the formulas for these measures, along with cumulative frequency tables and ogives, provide a complete system for analysing frequency distributions. The empirical relationship between the measures adds a valuable cross-check and connects the concepts. This chapter is highly scoring and practical, with applications in every field that uses data. A careful, stepwise approach to constructing tables and substituting formulas will secure full marks in this essential and rewarding unit of the Class 10 mathematics curriculum.