Interpreting and representing data

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शिक्षकों के लिए: Interpreting and representing data (Mathematics, Statistics) के लिए इस्तेमाल के लिए तैयार लेसन स्लाइड्स, रिवीज़न नोट्स — इन्हें अपने लेसन में इस्तेमाल करें, या टॉपिक को एक इंटरैक्टिव क्लास एक्टिविटी की तरह चलाएं जिसे आपके शिक्षार्थी लाइव गेम की तरह खेलें।

लेसन नोट्स स्लाइड्स प्रैक्टिस सवाल

लेसन नोट्स

Choosing the Right Chart

Categorical data (e.g. favourite colour) → bar charts, pie charts, or pictograms.
Ungrouped discrete numerical data (e.g. shoe size) → vertical line charts (also called bar-line graphs).
Time series data (e.g. temperature each month) → line graphs or time series graphs.
Always include a title, labels on axes, and a key for pictograms or multiple bars.
For bar charts, use equal bar widths and a gap between bars; the height shows frequency.
Pie charts show proportions of a whole, not frequencies – the total must be known.

Constructing Pie Charts

Each category's angle = (frequency ÷ total) × 360°.
Check that all angles add up to 360°.
Draw sector angles accurately with a protractor. Starting at 12 o’clock and working clockwise is convenient, but other starting positions are valid.
Label each sector with the category and percentage or angle.
Pie charts are best when comparing parts to a whole with a small number of categories.

Favourite colours of 60 students

Bar Charts and Pictograms

Bar charts have separate bars for each category; the height (or length) represents frequency.
Bars must have equal width and can be vertical or horizontal.
Pictograms use symbols to represent a set number of items (e.g. one symbol = 5 people).
Always include a key showing what one symbol represents.
When drawing pictograms, use half or quarter symbols if needed, but keep them consistent.

Vertical Line Charts

Used for ungrouped discrete data (e.g. number of siblings).
Draw a vertical line (or dot) above each value, with height equal to frequency.
The x-axis shows the data values, the y-axis shows frequency.
Use a numerical horizontal scale: vertical lines sit at the actual data values, so unequal value gaps have unequal spacing.
This chart is useful for showing the shape of the distribution.

Time Series Graphs

Plot data over time (e.g. months, years) on the x-axis.
Join the points with straight lines to show trends.
Look for trends (overall direction), seasonal patterns (regular fluctuations), and outliers.
Use a suitable scale so the graph is not misleading.
Time series graphs help compare changes over time for one or more variables.

Histograms (Higher Tier)

Histograms are for continuous data, often with unequal class widths.
The area of each bar represents frequency, not the height.
Calculate frequency density = frequency ÷ class width.
Plot frequency density on the y-axis and class boundaries on the x-axis.
Bars are drawn touching (no gaps) because data is continuous.
To find frequency from a bar: frequency = frequency density × class width.

Unequal-width classes

Cumulative Frequency Graphs (Higher Tier)

Cumulative frequency is the running total of frequencies up to each upper class boundary.
Plot cumulative frequency against the upper class boundary (not the midpoint).
Join points with a smooth curve (or straight lines).
To find the median, locate the value at ½ of the total frequency on the y-axis, then read across and down.
To find the lower quartile, use ¼ of total frequency; for upper quartile, use ¾.
The interquartile range = upper quartile − lower quartile.

Cumulative frequency

Box Plots (Higher Tier)

A box plot shows the minimum, lower quartile, median, upper quartile, and maximum.
Draw a box from lower to upper quartile, with a line at the median.
Extend whiskers to the minimum and maximum values (unless outliers are shown).
Box plots allow quick comparison of two or more distributions.
The length of the box shows the interquartile range (middle 50% of data).

A five-number summary

Comparing Distributions

Always compare both a measure of centre (mean, median, mode) and a measure of spread (range, interquartile range).
For example: 'The median for group A is higher, but the interquartile range is larger, so group A has more variability.'
When comparing, use the same type of chart for both sets of data.
Be careful with misleading graphs: non-zero axes, incorrect scales, or omitted keys can distort comparisons.

Common Mistakes to Avoid

In histograms, do not use frequency as the height when class widths are unequal – use frequency density.
Plot cumulative frequency at the upper class boundary, not the midpoint.
Do not read a pie chart as a frequency without knowing the total.
Do not compare distributions by centre alone – always include spread.
Always include labels, titles, and keys; check that axes start at zero unless there is a good reason not to.

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साइन अप करें — फ्री लॉग इन

प्रैक्टिस सवाल

फ्री प्रीव्यू — 64 में से 8 सवाल। सभी देखने के लिए साइन अप करें।

1.Which type of chart is most suitable for displaying categorical data?
Easy
- ABar chart
- BHistogram
- CLine graph
- DCumulative frequency graph
2.A pictogram uses symbols to represent frequencies.
Easy
True or false?
3.A pie chart is drawn for 90 students. The angle for the 'Football' category is 120°. How many students chose football?
Medium
- A30
- B40
- C45
- D60
4.Which displays can show the frequency of each separate, ungrouped discrete value? Select all that apply.
Medium
- AVertical line chart
- BBar chart
- CHistogram
- DPictogram
- ECumulative frequency graph
5.When drawing a pie chart, the angle for each category is calculated as (frequency / total) × 360°.
Easy
True or false?
6.Match each graphical representation with the type of data it is most suitable for.
Medium
- Line graph
- Histogram
- Pie chart
- Time series data
- Continuous data with unequal class widths
- Categorical data
7.A histogram has a 0–10 class of frequency density 5 and a 10–20 class of frequency density 3. What is the ratio of their frequencies?
Medium
- A5:3
- B3:5
- C1:1
- D10:9
8.A box plot shows a median of 30, lower quartile 20, upper quartile 45, and whiskers extending to 10 and 60. Which statement is correct?
Hard
- AThe interquartile range is 25.
- BThe range is 40.
- CThe median is 45.
- DThe lower quartile is 30.

और सवाल

9.Favourite colours among 60 students are Red 15, Blue 20 and Green 25. What is the angle of the Blue sector in a pie chart?
Medium
- A90°
- B120°
- C150°
- D60°
10.Arrange the following steps in the correct order to draw a pie chart from a frequency table.
Medium
- Calculate the angle for each category.
- Draw a circle and use a protractor to measure each angle.
- Add a title and key.
- Find the total frequency.
11.A cumulative frequency graph is used to estimate the median. Which value on the vertical axis corresponds to the median?
Hard
- ATotal frequency
- BHalf of the total frequency
- CQuarter of the total frequency
- DThree-quarters of the total frequency
12.A bar chart can be used for time series data.
Easy
True or false?
13.Which is an error when choosing bar heights for a histogram with unequal class widths?
Medium
- AUsing frequency density for the vertical axis
- BUsing frequency as the height of each bar
- CUsing frequency divided by class width
- DLabelling the axes clearly
14.Match each type of data with the most suitable graphical representation.
Easy
- Categorical data
- Ungrouped discrete numerical data
- Time series data
- Pie chart
- Vertical line chart
- Line graph
15.Arrange the following steps in the correct order to construct a histogram with unequal class widths.
Medium
- Calculate frequency density for each class
- Draw bars with heights equal to frequency density
- Identify the class boundaries
- Label the axes and give a title
16.Which are errors when constructing histograms or cumulative-frequency graphs? Select all that apply.
Medium
- AUsing frequency as the bar height
- BPlotting cumulative frequency at the midpoint
- CUsing frequency density as the bar height
- DUsing class width as the bar height
- EPlotting cumulative frequency at the upper class boundary
17.In a histogram with unequal class widths, the height of each bar represents the frequency density, not the frequency.
Easy
True or false?
18.A cumulative frequency graph is plotted. Which point should be used to plot the cumulative frequency for the class interval 10–20?
Medium
- A(10, cumulative frequency)
- B(15, cumulative frequency)
- C(20, cumulative frequency)
- D(20, 0)
19.From a cumulative-frequency graph, the lower quartile is 25 and the upper quartile is 55. What is the interquartile range?
Medium
- A20
- B30
- C40
- D80
20.Match each measure to the cumulative-frequency level used to estimate it.
Medium
- Median
- Lower quartile
- Upper quartile
- Value at 50% of the total frequency
- Value at 25% of the total frequency
- Value at 75% of the total frequency
21.Which of the following are appropriate for displaying time series data? (Select all that apply)
Medium
- ALine graph
- BPie chart
- CBar chart
- DVertical line chart
- EHistogram
22.Which of the following is the correct formula for frequency density?
Easy
- Afrequency / class width
- Bclass width / frequency
- Cfrequency × class width
- Dfrequency - class width
23.A class interval has a frequency of 12 and a class width of 3. What is the frequency density?
Medium
- A4
- B36
- C0.25
- D15
24.A cumulative frequency graph is plotted. For the class interval 10–20, which point should be used to plot the cumulative frequency?
Medium
- A(20, cumulative frequency)
- B(10, cumulative frequency)
- C(15, cumulative frequency)
- D(10, frequency)
25.In a pie chart, the angle of a sector is proportional to the frequency of that category.
Easy
True or false?
26.Which type of graph is most appropriate for displaying time series data?
Easy
- ALine graph
- BPie chart
- CPictogram
- DBar chart
27.A histogram has classes 0–10 and 10–20 with frequencies 20 and 30. What is the ratio of their bar heights?
Medium
- A2:3
- B1:1
- C1:2
- D3:2
28.A distribution has lower quartile 25 and interquartile range 20. What is its upper quartile?
Medium
- A45
- B55
- C25
- D15
29.Which of the following are appropriate when comparing two distributions? (Select all that apply)
Medium
- ACompare their medians
- BCompare their interquartile ranges
- CCompare their modes only
- DCompare their ranges
30.In a histogram with unequal class widths, the height of each bar is proportional to the frequency of the class.
Medium
True or false?
31.A cumulative frequency graph shows the following: lower quartile = 20, median = 30, upper quartile = 50. What is the interquartile range?
Hard
- A30
- B20
- C10
- D40
32.Match each graph with the type of data it is most suitable for.
Medium
- Pie chart
- Histogram
- Time series graph
- Categorical data
- Continuous data with unequal class intervals
- Data collected over time
33.A vertical line chart is suitable for displaying ungrouped discrete numerical data.
Easy
True or false?
34.Which type of chart is most suitable for displaying time series data?
Easy
- APie chart
- BLine graph
- CPictogram
- DVertical line chart
35.In a pie chart, the angle for the 'Red' category is 90°. If the total frequency is 200, how many items are in the 'Red' category?
Easy
- A50
- B90
- C200
- D25
36.Arrange the following steps in the correct order to construct a cumulative frequency graph.
Medium
- Plot cumulative frequency against the upper class boundary for each class.
- Calculate the cumulative frequency for each class by adding frequencies successively.
- Join the points with a smooth curve or straight lines.
- Draw axes with the upper class boundary on the horizontal axis and cumulative frequency on the vertical axis.
37.A single pie chart is a good way to show the chronological trend of a quantity over time.
Easy
True or false?
38.A distribution has upper quartile 52.5 and interquartile range 25. What is its lower quartile?
Hard
- A27.5
- B32.5
- C15
- D25
39.A histogram has a class interval 10–20 with a frequency of 24. What is the frequency density?
Medium
- A2.4
- B0.4
- C24
- D240
40.A cumulative frequency graph is drawn. The cumulative frequency for the class interval 10–20 is 15. Which point should be plotted for this class interval?
Medium
- A(20, 15)
- B(10, 15)
- C(15, 15)
- D(15, 20)
41.Complete the sentence.
Easy
In a histogram, the vertical axis represents ____ for unequal class widths.
42.In a histogram with class width 5 and frequency density 3, what is the frequency of that class?
Medium
43.From a cumulative frequency graph, the lower quartile is 20 and the upper quartile is 50. What is the interquartile range?
Medium
44.What is the formula for frequency density?
Medium
45.Favourite colours among 60 students are Red 15, Blue 20 and Green 25. What is the angle of the Red sector in a pie chart?
Easy
46.What measure of spread is calculated as the difference between the upper and lower quartiles?
Easy
47.A class interval has a frequency of 12 and a class width of 4. What is the frequency density?
Medium
48.Complete the sentence about comparing distributions.
Medium
When comparing two distributions, you should compare a measure of central tendency (such as the median) and a measure of ____.
49.State one reason why starting the vertical axis of a bar chart at a value other than zero can be misleading.
Medium
50.A frequency table shows that 30 students chose 'Red' out of a total of 120. What is the angle, in degrees, of the sector for 'Red' in a pie chart?
Easy
51.In a histogram, a class interval has a frequency of 36 and a class width of 6. What is the frequency density?
Medium
52.A cumulative frequency graph shows that the lower quartile is 18 and the upper quartile is 42. What is the interquartile range?
Medium
53.Explain why using frequency instead of frequency density for the height of bars in a histogram with unequal class widths would be misleading.
Medium
54.Explain why using frequency as the height of a bar in a histogram with unequal class widths is misleading, and describe the correct approach.
Hard
55.Explain why using frequency instead of frequency density for the height of bars in a histogram with unequal class widths would be misleading, and describe the correct approach.
Hard
56.Explain how to compare two numerical distributions using a measure of centre and a measure of spread. Why is comparing only their centres insufficient?
Hard
57.Explain why using frequency instead of frequency density for the height of bars in a histogram with unequal class widths is misleading, and describe the correct approach.
Hard
58.In a histogram with class width 5 and frequency density 3, what is the frequency of that class?
Medium
- A15
- B17
- C16
- D14
59.
From a cumulative frequency graph, the lower quartile is 20 and the upper quartile is 50. What is the interquartile range?
Quartile Value
Lower 20
Upper 50
Medium
- A30
- B27
- C36
- D33
60.Favourite colours among 60 students are Red 15, Blue 20 and Green 25. What is the angle of the Red sector in a pie chart?
Easy
- A108 °
- B81 °
- C90 °
- D99 °
61.A class interval has a frequency of 12 and a class width of 4. What is the frequency density?
Medium
- A2
- B4
- C5
- D3
62.A frequency table shows that 30 students chose 'Red' out of a total of 120. What is the angle, in degrees, of the sector for 'Red' in a pie chart?
Easy
- A81 °
- B108 °
- C90 °
- D99 °
63.In a histogram, a class interval has a frequency of 36 and a class width of 6. What is the frequency density?
Medium
- A7
- B6
- C5
- D8
64.A cumulative frequency graph shows that the lower quartile is 18 and the upper quartile is 42. What is the interquartile range?
Medium
- A28
- B26
- C22
- D24

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जल्द आ रहा है

Quartile	Value
Lower	20
Upper	50