Data Processing, Class 12 Practical Geography

What Data Processing is about

Key concepts

Central tendency and the three averages

NCERT section Measures of Central Tendency

Variables like rainfall, elevation, population density, education level or age vary from place to place. To summarise them we want one representative number. This value usually lies near the centre of the distribution, the point around which items tend to cluster.

• Mean: sum of all values divided by the number of observations; the simple arithmetic average.
• Median: the value that splits an ordered series into two equal halves; it depends on rank, not on the size of the extreme values. With an even number of items, take the average of the two middle values.
• Mode: the value that occurs most often.

Mean of ungrouped data: direct and indirect methods

NCERT section Computing Mean from Ungrouped Data

Direct method: Mean = (sum of x) / N.

Indirect method (coding): choose an assumed mean (A), subtract it from each value to get small deviations d = x - A, then Mean = A + (sum of d) / N. Useful when there are many observations or large numbers.

NCERT example (Table 2.1): normal rainfall of seven Malwa Plateau districts (mm):

District	d = x - 800	x
Indore	179	979
Dewas	283	1083
Dhar	33	833
Ratlam	96	896
Ujjain	91	891
Mandsaur	25	825
Shajapur	177	977
Sum	884	6484

• Direct: 6484 / 7 = 926.29 mm.
• Indirect (A = 800): 800 + 884 / 7 = 800 + 126.29 = 926.29 mm.

Both methods give the same answer.

Mean of grouped data

NCERT section Computing Mean from Grouped Data

In a frequency table individual values are lost, so each class is represented by its midpoint (x).

Direct method: Mean = (sum of f x) / N, where N = sum of f.

Indirect method: Mean = A + (sum of f d) / N, where A is the midpoint of the assumed-mean class (best chosen near the middle of the series) and d = midpoint - A. Add the positive and negative f d values separately and keep the sign of the difference. NCERT notes the indirect method works for equal and unequal class intervals. With step deviations u = d / i (i = class width), Mean = A + (sum of f u / N) x i.

NCERT example (Tables 2.2, 2.3): daily wage of 99 factory workers:

Wage (Rs)	Midpoint x	f	f x	d = x - 100	u = d / 20	f d	f u
50-70	60	10	600	-40	-2	-400	-20
70-90	80	20	1600	-20	-1	-400	-20
90-110	100	25	2500	0	0	0	0
110-130	120	35	4200	20	1	700	35
130-150	140	9	1260	40	2	360	18
Sum		99	10160			260	13

• Direct: 10,160 / 99 = Rs 102.6.
• Indirect: 100 + 260 / 99 = 100 + 2.6 = Rs 102.6.
• Step deviation: 100 + (13 / 99) x 20 = 100 + 2.6 = Rs 102.6.

Median

NCERT section Median

The median is a positional average: the point with an equal number of cases on each side. Symbol M.

Ungrouped data: arrange the values in order; the median is the value of the ((N + 1) / 2)th item.

• NCERT example: Himalayan peak heights 8,126, 8,611, 7,817, 8,172, 8,076, 8,848 and 8,598 m. N = 7, so (7 + 1) / 2 = 4th item. In ascending order the 4th value is 8,172 m (three peaks are lower and three higher). M = 8,172 m.

Grouped data: M = l + (i / f) x (N/2 - c), where l = lower limit of the median class, i = class interval, f = frequency of the median class, N = total frequency, c = cumulative frequency of the class before the median class.

NCERT example (Table 2.4):

Class	f	Cumulative F
50-60	3	3
60-70	7	10
70-80	11	21
80-90	16	37
90-100	8	45
100-110	5	50

Steps: N/2 = 50/2 = 25. The first cumulative frequency above 25 is 37, so the median class is 80-90; l = 80, f = 16, c = 21, i = 10.
M = 80 + (10 / 16) x (25 - 21) = 80 + 2.5 = 82.5.

Mode

NCERT section Mode

The mode is the value occurring most frequently; symbol Z or M0. It is used less widely than the mean and median, and a data set may have more than one mode. Arrange values in order first so repeats are easy to see.

• Unimodal: scores 61, 10, 88, 37, 61, 72, 55, 61, 46, 22 -> ordered 10, 22, 37, 46, 55, 61, 61, 61, 72, 88. Mode = 61 (only one value repeats most).
• Bimodal: scores 82, 11, 57, 82, 08, 11, 82, 95, 41, 11 -> ordered 08, 11, 11, 11, 41, 57, 82, 82, 82, 95. Both 11 and 82 occur three times.
• Trimodal: three values share the highest frequency.
• Multimodal: many values recur equally.
• Without mode: no value repeats.

Comparing mean, median and mode: normal and skewed curves

NCERT section Comparison of Mean, Median and Mode

• Normal curve: a bell-shaped, symmetrical frequency curve. Most observations lie around the middle, and their number falls off evenly towards both extremes. Many human traits, such as intelligence, personality scores and student achievement, are normally distributed. A normal curve can show high or low variability.
• In a normal distribution, mean, median and mode are the same value (100 in NCERT's Fig. 2.1). The highest frequency is in the middle, with exactly half the scores above and half below. Very high and very low scores are rare.
• In skewed (distorted) data, shown as positive and negative skew in Figs. 2.2 and 2.3, the three measures do not coincide, so the effect of skew must be considered when choosing an average.
• From the definitions: the median depends only on rank, so extreme values do not pull it; the mode marks the hump, the most frequent value; the mean uses every value.

Data Processing in real life

Each concept, met in an everyday situation. Read the situation, answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Data Processing: key facts

Mean, ungrouped, direct

$Mean=(sumofx)/N$

Malwa rainfall: 6484 / 7 = 926.29 mm.

Mean, ungrouped, indirect

$Mean=A+(sumofd)/N,d=x-A$

A = assumed mean (coding).

Mean, grouped, direct

$Mean=(sumoffx)/N$

x = class midpoint; N = sum of f.

Mean, grouped, indirect

$Mean=A+(sumoffd)/N;orA+(sumoffu/N)xi$

u = d / i; i = class width.

Median, ungrouped

Value of the ((N + 1) / 2)th item in the ordered series

Even N: average of the two middle items.

Median, grouped

$M=l+(i/f)x(N/2-c)$

c = cumulative frequency of the pre-median class.

Mode

Most frequent value (Z or M0)

Unimodal, bimodal, trimodal, multimodal or no mode.

Tricks to remember Data Processing

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Three averages

  "Mean = share equally, Median = middle seat, Mode = most popular."

  Use it when: Definitions.

• Grouped median formula

  "Lower limit, plus width over f, times (half N minus c before)."

  Use it when: M = l + (i/f)(N/2 - c).

• Normal curve

  "Bell is fair: all three averages share the top."

  Use it when: Mean = median = mode.

• Median position

  "N plus one, halved, is the median's home."

  Use it when: Ungrouped median.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.What are the three groups of statistical techniques named in the chapter?Show answer

   Measures of central tendency, of dispersion, and of relationship.

2. 2.What is the mean normal rainfall of the seven Malwa districts?Show answer

   926.29 mm.

3. 3.What is coding?Show answer

   Subtracting an assumed mean from each value to work with smaller numbers.

4. 4.Which item is the median of 7 ordered values?Show answer

   The 4th item.

5. 5.What is a series with two modes called?Show answer

   Bimodal.

6. 6.In a normal distribution, how are the mean, median and mode related?Show answer

   They are the same value.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you name the three groups of statistical measures?
• ☐Can you define mean, median and mode?
• ☐Can you compute the mean by direct and indirect methods?
• ☐Can you explain why both methods give the same mean?
• ☐Can you compute the grouped mean using midpoints?
• ☐Can you find the median of an odd and an even series?
• ☐Can you identify the median class and apply the grouped median formula?
• ☐Can you tell unimodal, bimodal and multimodal series apart?
• ☐Can you describe a normal curve?
• ☐Can you explain why the median is not affected by extreme values?

Data Processing: frequently asked questions