Rivalry2

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Original 2026-01-01

Modified 2026-02-28

1 Mean is the<a>average value</a>of a set of numbers. To find the mean, we add all the values and divide the sum by the total number of values. Depending on the data type, we use different methods to find the mean.

2 Mean for Ungrouped Data

3 Ungrouped data refers to raw data that has not been organized into groups, classes, or intervals. To find the mean, add all the values and divide the sum by the number of values.

4 \(\bar{x} = \frac{\sum x}{n} \)

5 For example, find the mean of 4, 6, 8, 10.

6 \({\bar x} = {{4 + 6 + 8 + 10} \over 4}\)

7 \(= {28 \over 4} = 7\)

8 Mean for Grouped DataGrouped data refers to data organized into groups or class intervals. The mean for grouped data can be calculated using three methods:

9 <ul><li>Calculating Mean Using the Direct Method</li>

10 <li>Calculating Mean Using Assumed Mean Method</li>

11 <li>Calculating Mean Using Step Deviation Method</li>

12 </ul>Calculating Mean Using the Direct MethodIn the direct method, the mean for grouped data is calculated by multiplying each class midpoint by its frequency, summing the products, and dividing by the total frequency.

13 \(\bar{x} = \frac{\sum f x}{\sum f} \)

14 For example, find the mean of the given data set.

15 Class IntervalFrequencyMidpoint (x)fx0 - 10 4 5 20 10 - 20 6 15 90 20 - 30 5 25 125\(Σfx = (4 × 5) + (6 × 15) + (5 × 25) = 235 \)

16 \(Σf = 4 + 6 + 5 = 15 \)

17 \(\bar{x} = \frac{\sum f x}{\sum f} \)

18 \(\bar x = {235 \over 15}\)

19 = 15.67.

20 Calculating Mean Using Assumed Mean Method

21 The assumed mean method is used when the sample size is large. We take a mean (A) and calculate deviations from it.

22 Here, \( \bar{x} = A + \frac{\sum f d}{\sum f} \)

23 where, d = x - A

24 For example,

25 xfd = x - A10 2 -10 20 3 0 30 5 10 Let’s assume A = 20

26 Then,

27 \(Σfd = (-10 × 2) + (0 × 3) + (10 × 5) \)

28 \(= -20 + 0 + 50 \)

29 = 30

30 \({\bar x} = 20 + {30 \over 10}\)

31 \(= 20 + 3 = 23\)

32 Calculating Mean Using Step Deviation MethodThe<a>step deviation</a>method is used when class intervals are equal and values are significant. The formula to find the mean using the step deviation method is:

33 \(\bar{x} = A + h \frac{\sum f _u}{\sum f} \)

34 where A = assumed mean

35 h = class width

36 \(u = \frac{x - A}{h} \)

37 For example,

38 Class IntervalfMidpoint(x)u\(f_u\)10-20 5 15 -1 -5 20-30 7 25 0 0 30-40 8 35 1 8Here, A = 25

39 h = 10

40 \(Σf_u = 3 \)

41 \(Σf = 20 \)

42 Substitute the values in the<a>equation</a>:

43 \(\bar{x} = 25 + 10 ( \frac{30}{20}) \)

44 \({\bar x}= 25 + 1.5 = 26.5 \)

45 Mean of Negative Numbers

46 The method for finding the mean remains the same even when the numbers are negative. This means the mean of negative values is simply the sum of all the observations divided by the total number of observations.

47 For example, -5, 10, -3

48 \({\bar x} = {{-5 + 10 - 3} \over 3}\)

49 \(= {2 \over 3}\)

50 = 0.667.