Understanding Data

Overview

Learning outcomes

• Students will be able to define data and explain how processing turns data into information.
• Students will be able to give examples of data being used for decisions in everyday life.
• Students will be able to classify data as structured or unstructured and explain metadata with examples.
• Students will be able to describe data collection, data storage and the input-process-output cycle.
• Students will be able to calculate mean, median and mode of a data set and choose a suitable measure.
• Students will be able to calculate range and standard deviation and interpret what they say about spread.
• Students will be able to write short Python programs that compute these measures from a list.

Lesson plan

Notes

runs = [45, 12, 78, 45, 0, 102, 33, 45, 60]   # runs in 9 innings
total = sum(runs)
n = len(runs)
mean = total / n
print("Total runs:", total)
print("Innings   :", n)
print("Mean runs :", round(mean, 2))

Total runs: 420
Innings   : 9
Mean runs : 46.67

runs = [45, 12, 78, 45, 0, 102, 33, 45, 60]
s = sorted(runs)
n = len(s)
print("Sorted:", s)
if n % 2 == 1:
    median = s[n // 2]
else:
    median = (s[n // 2 - 1] + s[n // 2]) / 2
print("Median:", median)

heights = [150, 162, 158, 171, 149, 166]   # even count
h = sorted(heights)
print("Sorted heights:", h)
print("Median height :", (h[2] + h[3]) / 2)

Sorted: [0, 12, 33, 45, 45, 45, 60, 78, 102]
Median: 45
Sorted heights: [149, 150, 158, 162, 166, 171]
Median height : 160.0

runs = [45, 12, 78, 45, 0, 102, 33, 45, 60]
freq = {}
for r in runs:
    freq[r] = freq.get(r, 0) + 1
print("Frequency table:", freq)
highest = max(freq.values())
modes = [k for k in freq if freq[k] == highest]
print("Mode:", modes, "occurs", highest, "times")

houses = ["Tagore", "Raman", "Tagore", "Ashoka", "Raman", "Tagore"]
print("Most common house:", max(houses, key=houses.count))

Frequency table: {45: 3, 12: 1, 78: 1, 0: 1, 102: 1, 33: 1, 60: 1}
Mode: [45] occurs 3 times
Most common house: Tagore

pocket = [500, 600, 550, 650, 600, 5000]   # one friend gets Rs 5000
mean = sum(pocket) / len(pocket)
s = sorted(pocket)
median = (s[2] + s[3]) / 2
print("Mean  :", round(mean, 2))
print("Median:", median)

without = [500, 600, 550, 650, 600]
print("Mean without outlier:", sum(without) / len(without))

Mean  : 1316.67
Median: 600.0
Mean without outlier: 580.0

temp_delhi = [41, 43, 44, 42, 45, 46, 44]   # max temperature in a week (deg C)
temp_shimla = [18, 25, 21, 30, 16, 27, 20]
print("Delhi range :", max(temp_delhi) - min(temp_delhi))
print("Shimla range:", max(temp_shimla) - min(temp_shimla))

Delhi range : 5
Shimla range: 14

marks = [6, 8, 10, 12, 14]          # marks out of 20
n = len(marks)
mean = sum(marks) / n
sq = [(x - mean) ** 2 for x in marks]
print("Mean:", mean)
print("Squared deviations:", sq)
var = sum(sq) / n
sd = var ** 0.5
print("Variance:", var)
print("Standard deviation:", round(sd, 2))

Mean: 10.0
Squared deviations: [16.0, 4.0, 0.0, 4.0, 16.0]
Variance: 8.0
Standard deviation: 2.83

import statistics as st

runs = [45, 12, 78, 45, 0, 102, 33, 45, 60]
print("mean   :", round(st.mean(runs), 2))
print("median :", st.median(runs))
print("mode   :", st.mode(runs))
print("range  :", max(runs) - min(runs))
print("pstdev :", round(st.pstdev(runs), 2))   # divides by n (as in this chapter)
print("stdev  :", round(st.stdev(runs), 2))    # divides by n-1 (sample)

mean   : 46.67
median : 45
mode   : 45
range  : 102
pstdev : 29.43
stdev  : 31.22

import pandas as pd

marks = pd.Series([67, 82, 91, 45, 82, 74, 58, 82, 99, 60])
print("Mean   :", marks.mean())
print("Median :", marks.median())
print("Mode   :", marks.mode().tolist())
print("Range  :", marks.max() - marks.min())
print("Std    :", round(marks.std(ddof=0), 2))

Mean   : 74.0
Median : 78.0
Mode   : [82]
Range  : 54
Std    : 15.65

Key terms

Data

Raw facts, figures, symbols, images or sounds that describe something.

Information

Data that has been processed so that it is meaningful and useful.

Structured data

Data organised in a fixed format of rows (records) and columns (attributes).

Unstructured data

Data with no fixed row-column format, such as text, images, audio and video.

Attribute

A column in structured data; one characteristic recorded for every record.

Metadata

Data about data, for example the size, type and date of a photo.

Data processing

Converting raw data into information through input, processing and output.

Mean

Sum of all values divided by the number of values; the average.

Median

The middle value of the data after sorting.

Mode

The value that occurs most often in the data.

Outlier

A value that is unusually large or small compared with the rest.

Range

Maximum value minus minimum value.

Standard deviation

Square root of the average squared distance of values from the mean.

Variance

Average of the squared deviations from the mean; the square of standard deviation.

MCQs

Tap an option to check it, or use Show answers above to reveal them all.

1. Which of these is an example of unstructured data?

   1. A railway reservation chart
   2. A marks register in a spreadsheet
   3. A voice note on WhatsApp
   4. A fee receipt table
   Answer

   (c) A voice note on WhatsApp

   A voice note has no row-column format. The other three are tables.

2. Processed and meaningful data is called:

   1. Information
   2. Metadata
   3. Datum
   4. Attribute
   Answer

   (a) Information

   Processing converts raw data into information.

3. The size, resolution and date-taken of a photo are examples of:

   1. Structured data
   2. Outliers
   3. Raw data
   4. Metadata
   Answer

   (d) Metadata

   Metadata is data that describes other data.

4. The mean of 4, 8, 6, 10 and 12 is:

   1. 6
   2. 8
   3. 10
   4. 40
   Answer

   (b) 8

   (4 + 8 + 6 + 10 + 12) / 5 = 40 / 5 = 8.

5. The median of 15, 3, 9, 21, 6, 12 is:

   1. 9
   2. 12
   3. 10.5
   4. 11
   Answer

   (c) 10.5

   Sorted: 3, 6, 9, 12, 15, 21. Even count, so median = (9 + 12) / 2 = 10.5.

6. Which measure can be found for non-numeric data such as favourite sport?

   1. Mean
   2. Median
   3. Range
   4. Mode
   Answer

   (d) Mode

   Mode only needs counting how often each value occurs.

7. Which measure is most affected by an outlier?

   1. Mean
   2. Median
   3. Mode
   4. None of these
   Answer

   (a) Mean

   The mean adds every value, so one extreme value pulls it strongly.

8. The range of 22, 35, 18, 40, 27 is:

   1. 18
   2. 22
   3. 40
   4. 13
   Answer

   (b) 22

   40 − 18 = 22.

9. A small standard deviation tells us that the values are:

   1. All equal to zero
   2. Widely spread out
   3. Close to the mean
   4. Mostly outliers
   Answer

   (c) Close to the mean

   Standard deviation measures spread around the mean; small means tightly grouped.

10. The data set 7, 3, 9, 3, 7, 5 has:

    1. No mode
    2. Two modes, 3 and 7
    3. One mode, 7
    4. One mode, 5
    Answer

    (b) Two modes, 3 and 7

    Both 3 and 7 occur twice, more than any other value.

11. In the input-process-output cycle, generating an admit card is part of:

    1. Input
    2. Processing
    3. Output
    4. Storage
    Answer

    (c) Output

    The admit card is the result produced after processing the form data.

12. If the variance of a data set is 25, its standard deviation is:

    1. 5
    2. 625
    3. 12.5
    4. 25
    Answer

    (a) 5

    Standard deviation is the positive square root of variance: √25 = 5.

Assertion–Reason

1. Assertion (A): The median is often a better measure of centre than the mean for data on family incomes in a town.

   Reason (R): The median is not much affected by a few extremely large values.

   Answer

   (A)

   Incomes usually contain a few very large values (outliers). The median resists them, which is exactly why it is preferred.

2. Assertion (A): Mode can be calculated for the list of house names of students in a class.

   Reason (R): Mode is the value with the highest frequency, and frequency can be counted for any kind of value.

   Answer

   (A)

   Both are true and R explains why mode works for non-numeric data.

3. Assertion (A): Range uses every value in the data set.

   Reason (R): Range is the difference between the maximum and minimum values.

   Answer

   (D)

   Range uses only the two extreme values, so A is false. R is the correct definition.

4. Assertion (A): An email is an example of unstructured data.

   Reason (R): Unstructured data cannot be stored on a computer.

   Answer

   (C)

   An email has no fixed row-column format, so A is true. Unstructured data is stored on computers all the time, so R is false.

Short answer

1. 2 marks

   Differentiate between data and information with one example.

   Answer

   Data is raw, unprocessed facts, for example the marks of each student in a class. Information is processed, meaningful data, for example "the class average in IP is 62 and 5 students scored above 90". Information is obtained by processing data.

2. 2 marks

   What is structured data? Give two examples.

   Answer

   Structured data is organised in a fixed format of rows and columns, where each column is an attribute and each row is a record. Examples: a class marks register in a spreadsheet; a shop's stock list with ItemCode, ItemName, Price and Quantity.

3. 2 marks

   What is metadata? Give the metadata of an email.

   Answer

   Metadata is data about data; it describes other data. The metadata of an email includes the sender, recipients, subject, date and time sent, and names and sizes of attachments.

4. 2 marks

   Find the mean and median of 12, 7, 19, 7, 15.

   Answer

   Mean = (12 + 7 + 19 + 7 + 15) / 5 = 60 / 5 = 12.

   Sorted: 7, 7, 12, 15, 19. Odd count, so median = 3rd value = 12.

5. 2 marks

   What is an outlier? How does it affect the mean?

   Answer

   An outlier is a value that is unusually large or small compared with the rest of the data. Because the mean adds up every value, an outlier pulls the mean towards itself, so the mean no longer represents a typical value.

6. 2 marks

   Find the mode and range of 5, 9, 2, 9, 4, 9, 2.

   Answer

   9 occurs three times, more than any other value, so mode = 9.

   Range = maximum − minimum = 9 − 2 = 7.

7. 2 marks

   Why is standard deviation considered a better measure of spread than range?

   Answer

   Range uses only the maximum and minimum values, so one extreme value can change it a lot and it ignores how the other values are spread. Standard deviation uses every value's distance from the mean, so it gives a fuller picture of spread.

8. 2 marks

   Write the steps of the input-process-output cycle for booking a movie ticket online.

   Answer

   • Input: movie, show time, number of seats, seat choice, payment details.
   • Process: check seat availability, block the seats, verify payment, update the seat map.
   • Output: e-ticket with booking ID and seat numbers, sent by SMS/email.

Long answer

1. 3 marks

   The heights (in cm) of 8 students are 152, 160, 148, 165, 160, 155, 170, 160. Find the mean, median, mode and range.

   Answer

   Sum = 152 + 160 + 148 + 165 + 160 + 155 + 170 + 160 = 1270. Mean = 1270 / 8 = 158.75 cm.

   Sorted: 148, 152, 155, 160, 160, 160, 165, 170. Even count, so median = (4th + 5th) / 2 = (160 + 160) / 2 = 160 cm.

   160 occurs three times, so mode = 160 cm.

   Range = 170 − 148 = 22 cm.

2. 3 marks

   Calculate the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9. Show each step.

   Answer

   1. Mean = (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 40 / 8 = 5.
   2. Deviations from mean: −3, −1, −1, −1, 0, 0, 2, 4.
   3. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16. Sum = 32.
   4. Variance = 32 / 8 = 4.
   5. Standard deviation = √4 = 2.
3. 5 marks

   Explain with examples any four areas where data helps in decision making.

   Answer

   • Banking: account data is updated after each ATM withdrawal; past repayment data is used to approve or reject a loan.
   • Weather forecasting: the IMD analyses satellite and sensor data to issue cyclone and heavy-rain warnings so that people can be moved to safety.
   • Railways and airlines: booking data drives dynamic pricing; fares rise as seats fill up on popular dates.
   • Retail: a supermarket studies which items are bought together and places them near each other to increase sales.
   • Government planning: census data on population and literacy helps decide where to open new schools and hospitals.

   (Any four, with a clear example each.)

4. 5 marks

   Two batsmen scored the following runs in five matches. A: 40, 42, 38, 41, 39. B: 5, 90, 10, 80, 15. (a) Find the mean for both. (b) Find the range for both. (c) Which batsman is more consistent? Justify using spread.

   Answer

   (a) Mean of A = 200 / 5 = 40. Mean of B = 200 / 5 = 40. Both have the same mean.

   (b) Range of A = 42 − 38 = 4. Range of B = 90 − 5 = 85.

   (c) Batsman A is more consistent. Both average 40 runs, but A's scores lie very close to 40 (small range, and so a small standard deviation) while B's scores swing between very low and very high. Same mean does not mean same behaviour; the measure of spread shows the difference.

Case-based

Practicals

# Summary of Class XI-A IP marks (out of 70)
marks = [56, 61, 48, 70, 61, 39, 65, 61, 52, 58]

n = len(marks)
mean = sum(marks) / n

s = sorted(marks)
if n % 2 == 1:
    median = s[n // 2]
else:
    median = (s[n // 2 - 1] + s[n // 2]) / 2

freq = {}
for m in marks:
    freq[m] = freq.get(m, 0) + 1
top = max(freq.values())
mode = [k for k in freq if freq[k] == top]

rng = max(marks) - min(marks)
sd = (sum((x - mean) ** 2 for x in marks) / n) ** 0.5

print("Number of students:", n)
print("Mean     :", mean)
print("Median   :", median)
print("Mode     :", mode)
print("Range    :", rng)
print("Std. dev :", round(sd, 2))

Number of students: 10
Mean     : 57.1
Median   : 59.5
Mode     : [61]
Range    : 31
Std. dev : 8.44

# Compare two bowlers: runs given in their last 6 overs
bowler_a = [6, 7, 6, 8, 7, 8]
bowler_b = [2, 14, 1, 12, 3, 10]

def mean(d):
    return sum(d) / len(d)

def sd(d):
    m = mean(d)
    return (sum((x - m) ** 2 for x in d) / len(d)) ** 0.5

for name, data in [("Bowler A", bowler_a), ("Bowler B", bowler_b)]:
    print(name, "mean =", round(mean(data), 2),
          "range =", max(data) - min(data),
          "sd =", round(sd(data), 2))

Bowler A mean = 7.0 range = 2 sd = 0.82
Bowler B mean = 7.0 range = 13 sd = 5.16

# Classify each item as structured or unstructured data
items = {
    "Fee receipt table (AdmNo, Name, Month, Amount)": "structured",
    "Voice note sent on a class group": "unstructured",
    "Railway reservation chart": "structured",
    "Photo of the school annual day": "unstructured",
    "Attendance register in a spreadsheet": "structured",
    "Principal's speech video": "unstructured",
}
s = [k for k, v in items.items() if v == "structured"]
u = [k for k, v in items.items() if v == "unstructured"]
print("Structured  :", len(s))
for k in s:
    print("  -", k)
print("Unstructured:", len(u))
for k in u:
    print("  -", k)

Structured  : 3
  - Fee receipt table (AdmNo, Name, Month, Amount)
  - Railway reservation chart
  - Attendance register in a spreadsheet
Unstructured: 3
  - Voice note sent on a class group
  - Photo of the school annual day
  - Principal's speech video

Common mistakes

• Finding the median without sorting the data first. Correction: always write the sorted list, then pick the middle value (or average the two middle values when n is even).

• Saying a data set always has exactly one mode. Correction: it may have no mode (all values occur once) or more than one mode.

• Using the mean to describe data with an outlier, such as one very large salary. Correction: use the median, or check and remove the outlier first.

• Forgetting to take the square root, and reporting the variance as the standard deviation. Correction: σ = √variance.

• Calling a photo or email "structured" because it is stored on a computer. Correction: structured means organised in rows and columns, not merely digital.

• Mixing up data and information. Correction: data is raw input; information is the processed, meaningful result.