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Class 10 — Mathematics (NCERT)

Statistics: Mean, Median, Mode of Grouped Data and Ogives

Class 10

  • ✓By the end of this lesson students will be able to calculate the mean of grouped data using direct, assumed mean, and step-deviation methods.
  • ✓By the end of this lesson students will be able to calculate the mode of grouped data.
  • ✓By the end of this lesson students will be able to calculate the median of grouped data.
  • ✓By the end of this lesson students will be able to construct 'less than' and 'more than' type cumulative frequency curves (ogives).
  • ✓By the end of this lesson students will be able to determine the median of grouped data graphically using ogives.

Key concepts

Mean of Grouped Data

The mean of grouped data is a measure of central tendency that represents the average value of the data. Since individual observations are not known, we use the class mark (mid-point) of each class interval to represent the observations within that class. There are three methods to calculate the mean of grouped data.

Direct Method for Mean

This method is used when the frequencies and class marks are small. It involves finding the product of each frequency and its corresponding class mark, summing these products, and then dividing by the total frequency.

Mean (x̄) = Σ(f_i * x_i) / Σf_i\nwhere f_i is the frequency of the i-th class and x_i is the class mark of the i-th class.
Assumed Mean Method for Mean

This method simplifies calculations when the class marks and frequencies are large. We assume a mean (A) from one of the class marks, usually the middle one, and then calculate deviations from this assumed mean.

Mean (x̄) = A + [Σ(f_i * d_i) / Σf_i]\nwhere A is the assumed mean, f_i is the frequency, and d_i = x_i - A is the deviation of the i-th class mark from the assumed mean.
Step-Deviation Method for Mean

This method is a further simplification of the assumed mean method, especially useful when class sizes are uniform. It involves dividing the deviations by the class size (h) to get step-deviations, making the numbers smaller and easier to calculate.

Mean (x̄) = A + [Σ(f_i * u_i) / Σf_i] * h\nwhere A is the assumed mean, f_i is the frequency, u_i = (x_i - A) / h is the step-deviation, and h is the class size.
Mode of Grouped Data

The mode is the value that appears most frequently in a data set. For grouped data, we first identify the 'modal class', which is the class interval with the highest frequency. The mode lies within this modal class.

Mode = L + [(f_1 - f_0) / (2f_1 - f_0 - f_2)] * h\nwhere:\nL = lower limit of the modal class\nh = size of the class interval\nf_1 = frequency of the modal class\nf_0 = frequency of the class preceding the modal class\nf_2 = frequency of the class succeeding the modal class
Median of Grouped Data

The median is the middle value of a data set when it is arranged in ascending or descending order. For grouped data, we first find the 'median class', which is the class interval whose cumulative frequency is just greater than or equal to N/2 (where N is the total frequency). The median lies within this median class.

Median = L + [(N/2 - C_f) / f] * h\nwhere:\nL = lower limit of the median class\nN = total frequency (Σf_i)\nC_f = cumulative frequency of the class preceding the median class\nf = frequency of the median class\nh = size of the class interval
Cumulative Frequency (C.F.)

Cumulative frequency is the running total of frequencies. It helps in determining the median and constructing ogives. There are two types:\n1. 'Less than' type: The cumulative frequency of a class is the sum of frequencies of that class and all classes preceding it. It is associated with the upper limit of the class interval.\n2. 'More than' type: The cumulative frequency of a class is the sum of frequencies of that class and all classes succeeding it. It is associated with the lower limit of the class interval.

Ogives (Cumulative Frequency Curves)

An ogive is a graphical representation of cumulative frequency distribution. It is used to determine the median graphically. There are two types of ogives:\n1. 'Less than' ogive: Points are plotted with upper class limits on the x-axis and corresponding 'less than' cumulative frequencies on the y-axis. These points are joined by a smooth curve.\n2. 'More than' ogive: Points are plotted with lower class limits on the x-axis and corresponding 'more than' cumulative frequencies on the y-axis. These points are joined by a smooth curve.\nThe median can be found by locating N/2 on the y-axis, drawing a horizontal line to the ogive, and then a vertical line to the x-axis. The x-coordinate of this point is the median. If both ogives are drawn on the same graph, their intersection point's x-coordinate gives the median.

Key facts to remember

  • 1The mean, median, and mode are measures of central tendency.
  • 2Mean is the average, median is the middle value, and mode is the most frequent value.
  • 3For grouped data, class marks are used to calculate the mean.
  • 4The modal class is the class with the highest frequency; the median class is where the cumulative frequency first exceeds N/2.
  • 5The empirical relationship between the three measures is: Mode ≈ 3 Median - 2 Mean.
  • 6Cumulative frequency curves (ogives) are graphical representations used to find the median graphically.
  • 7'Less than' ogive uses upper class limits and 'less than' cumulative frequencies. 'More than' ogive uses lower class limits and 'more than' cumulative frequencies.

Worked examples

Example 1

The following distribution shows the daily pocket allowance of children of a locality. Find the mean daily pocket allowance using the step-deviation method.

IFirst, we prepare the frequency distribution table with class marks (x_i), deviations (d_i), step-deviations (u_i), and products (f_i * u_i).
IILet the assumed mean (A) be 18 (mid-point of class 17-19). The class size (h) is 2.
III| Daily Pocket Allowance (₹) | Number of Children (f_i) | Class Mark (x_i) | d_i = x_i - A = x_i - 18 | u_i = d_i / h = d_i / 2 | f_i * u_i |
IV| :------------------------- | :----------------------- | :--------------- | :------------------------- | :---------------------- | :---------- |
V| 11-13 | 7 | 12 | -6 | -3 | -21 |
VI| 13-15 | 6 | 14 | -4 | -2 | -12 |
VII| 15-17 | 9 | 16 | -2 | -1 | -9 |
VIII| 17-19 | 13 | 18 | 0 | 0 | 0 |
9| 19-21 | 20 | 20 | 2 | 1 | 20 |
10| 21-23 | 5 | 22 | 4 | 2 | 10 |
11| 23-25 | 4 | 24 | 6 | 3 | 12 |
12| **Total** | **Σf_i = 64** | | | | **Σf_i * u_i = 0** |
13Now, we apply the step-deviation formula:
14Mean (x̄) = A + [Σ(f_i * u_i) / Σf_i] * h
15x̄ = 18 + [0 / 64] * 2
16x̄ = 18 + 0 * 2
17x̄ = 18 + 0
18x̄ = 18

Answer

The mean daily pocket allowance is ₹18.

The step-deviation method simplifies calculations, especially when class intervals are uniform and frequencies are large.

Example 2

The following frequency distribution gives the information about the observed life times (in hours) of 225 electrical components. Find the median life time of the components.

IFirst, we prepare the cumulative frequency table.
II| Life Time (in hours) | Frequency (f_i) | Cumulative Frequency (C.F.) |
III| :------------------- | :-------------- | :-------------------------- |
IV| 0-20 | 10 | 10 |
V| 20-40 | 35 | 10 + 35 = 45 |
VI| 40-60 | 52 | 45 + 52 = 97 |
VII| 60-80 | 61 | 97 + 61 = 158 |
VIII| 80-100 | 38 | 158 + 38 = 196 |
9| 100-120 | 29 | 196 + 29 = 225 |
10Here, N = Σf_i = 225. So, N/2 = 225/2 = 112.5.
11The cumulative frequency just greater than 112.5 is 158, which corresponds to the class interval 60-80. Thus, the median class is 60-80.
12From the median class, we have:
13L = lower limit = 60
14f = frequency of the median class = 61
15C_f = cumulative frequency of the class preceding the median class = 97
16h = class size = 20
17Now, we apply the median formula:
18Median = L + [(N/2 - C_f) / f] * h
19Median = 60 + [(112.5 - 97) / 61] * 20
20Median = 60 + [15.5 / 61] * 20
21Median = 60 + [0.254098...] * 20
22Median = 60 + 5.08196...
23Median ≈ 65.08

Answer

The median life time of the components is approximately 65.08 hours.

Always ensure the data is in continuous class intervals before calculating the median. If not, make it continuous.

Example 3

The following data gives the information on the observed life times (in hours) of 225 electrical components. Find the modal life time of the components.

IFirst, we identify the modal class. The class with the highest frequency is the modal class.
II| Life Time (in hours) | Frequency (f_i) |
III| :------------------- | :-------------- |
IV| 0-20 | 10 |
V| 20-40 | 35 |
VI| 40-60 | 52 |
VII| 60-80 | 61 |
VIII| 80-100 | 38 |
9| 100-120 | 29 |
10The highest frequency is 61, which corresponds to the class interval 60-80. Therefore, the modal class is 60-80.
11From the modal class, we have:
12L = lower limit of the modal class = 60
13h = class size = 20
14f_1 = frequency of the modal class = 61
15f_0 = frequency of the class preceding the modal class = 52
16f_2 = frequency of the class succeeding the modal class = 38
17Now, we apply the mode formula:
18Mode = L + [(f_1 - f_0) / (2f_1 - f_0 - f_2)] * h
19Mode = 60 + [(61 - 52) / (2*61 - 52 - 38)] * 20
20Mode = 60 + [9 / (122 - 52 - 38)] * 20
21Mode = 60 + [9 / (122 - 90)] * 20
22Mode = 60 + [9 / 32] * 20
23Mode = 60 + [0.28125] * 20
24Mode = 60 + 5.625
25Mode = 65.625

Answer

The modal life time of the components is 65.625 hours.

The mode is the value with the highest frequency. In grouped data, it is an estimate within the modal class.

Example 4

The following table gives the production yield per hectare of wheat of 100 farms of a village. Draw both 'less than' and 'more than' type ogives and find the median from the graph.

IFirst, we prepare the cumulative frequency tables for 'less than' and 'more than' types.
II**'Less than' type cumulative frequency distribution:**
III| Production Yield (kg/ha) | Number of Farms (f_i) | Upper Class Limit | Less than C.F. |
IV| :----------------------- | :-------------------- | :---------------- | :------------- |
V| 50-55 | 2 | 55 | 2 |
VI| 55-60 | 8 | 60 | 2 + 8 = 10 |
VII| 60-65 | 12 | 65 | 10 + 12 = 22 |
VIII| 65-70 | 24 | 70 | 22 + 24 = 46 |
9| 70-75 | 38 | 75 | 46 + 38 = 84 |
10| 75-80 | 16 | 80 | 84 + 16 = 100 |
11Points for 'less than' ogive: (55, 2), (60, 10), (65, 22), (70, 46), (75, 84), (80, 100).
12**'More than' type cumulative frequency distribution:**
13| Production Yield (kg/ha) | Number of Farms (f_i) | Lower Class Limit | More than C.F. |
14| :----------------------- | :-------------------- | :---------------- | :------------- |
15| 50-55 | 2 | 50 | 100 |
16| 55-60 | 8 | 55 | 100 - 2 = 98 |
17| 60-65 | 12 | 60 | 98 - 8 = 90 |
18| 65-70 | 24 | 65 | 90 - 12 = 78 |
19| 70-75 | 38 | 70 | 78 - 24 = 54 |
20| 75-80 | 16 | 75 | 54 - 38 = 16 |
21Points for 'more than' ogive: (50, 100), (55, 98), (60, 90), (65, 78), (70, 54), (75, 16).
22Now, plot these points on a graph paper. Take 'Production Yield' on the x-axis and 'Cumulative Frequency' on the y-axis. Draw smooth curves joining the points for both types of ogives.
23To find the median graphically: The total number of farms (N) is 100. So, N/2 = 100/2 = 50.
24Locate 50 on the y-axis. Draw a horizontal line from y=50 to intersect both ogives. From the point of intersection, draw a vertical line to the x-axis. The x-coordinate where this vertical line meets the x-axis is the median.
25Alternatively, the x-coordinate of the intersection point of the 'less than' ogive and the 'more than' ogive gives the median.
26Upon plotting and observing the intersection point, the x-coordinate will be approximately 70.5.

Answer

The median production yield per hectare, found graphically from the ogives, is approximately 70.5 kg/ha.

For drawing ogives, always use graph paper for accuracy. The intersection point of the two ogives gives the median.

Common mistakes

  • ✗Incorrectly calculating class marks (x_i) or cumulative frequencies (C.F.).
  • ✗Confusing the lower limit (L) with the upper limit when applying formulas for median or mode.
  • ✗Using the frequency of the preceding class instead of the frequency of the median/modal class (f) in the formulas, or vice-versa.
  • ✗Errors in identifying f_0, f_1, and f_2 for the mode formula.
  • ✗Plotting points incorrectly for ogives (e.g., using lower limit for 'less than' ogive or upper limit for 'more than' ogive).

Exam tips

  • ★Always read the question carefully to determine which measure of central tendency is required and which method (for mean) to use.
  • ★Draw clear and well-labelled tables for calculations, especially for mean, median, and ogives.
  • ★Use graph paper for drawing ogives to ensure accuracy in plotting points and finding the median.
  • ★Double-check all calculations, especially when dealing with large numbers or fractions, to avoid arithmetic errors.

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