Statistical dataను textual లేదా tabular formలో మాత్రమే కాకుండా diagrams and graphs ద్వారా కూడా present చేయవచ్చు. Numbersలో ఉన్న informationను visual formలో చూపడం వల్ల dataను త్వరగా understand చేయడం, compare చేయడం మరియు trendsను identify చేయడం సులభమవుతుంది.
Diagrammatic Presentation
Geometric Diagrams
Simple Bar Diagram
Multiple Bar Diagram
Component Bar Diagram
Pie Diagram / Pie Chart
Frequency Diagrams
Histogram
Unequal Class Intervals & Frequency Density
Frequency Polygon
Frequency Curve
Ogives – Less Than & More Than
Median from Ogives
Arithmetic Line Graph / Time Series Graph
Bar Diagram vs Histogram
Numerical Problems & Shortcuts
Numerical dataను diagrams ద్వారా visually represent చేయడాన్ని Diagrammatic Presentation of Data అంటారు.
Tablesతో పోలిస్తే diagrams కొన్ని సందర్భాల్లో less precise కావచ్చు, కానీ informationను quickly communicate చేయడంలో చాలా effective.
Bar Diagram + Pie Diagram
Histogram + Frequency Polygon + Frequency Curve + Ogive
Time Series Graph
Important geometric diagrams:
2. Pie Diagram / Pie Chart
Bar diagramలో each categoryకు ఒక rectangular bar draw చేస్తారు. Bars సాధారణంగా equal width కలిగి, వాటి మధ్య equal spacing ఉంటుంది.
Bar యొక్క height లేదా length corresponding value యొక్క magnitudeను represent చేస్తుంది.
Production:
2022 → 40 tonnes
2023 → 60 tonnes
2024 → 80 tonnes
The bar for 2024 will be the tallest.
• Bars are separated by spaces.
• Height/length represents magnitude.
• Bars normally start from a common baseline.
• Categories can be compared visually.
• It can be used for frequency as well as non-frequency data.
2. Multiple Bar Diagram
3. Component Bar Diagram
ఒకే variable/characteristicకు సంబంధించిన valuesను compare చేయడానికి Simple Bar Diagram ఉపయోగిస్తారు.
| Year | Production |
|---|---|
| 2022 | 50 |
| 2023 | 70 |
| 2024 | 90 |
One set of values across categories → Simple Bar Diagram
Two or more related sets of dataను same categoriesపై compare చేయడానికి Multiple Bar Diagram ఉపయోగిస్తారు.
Income vs Expenditure
Imports vs Exports
Male vs Female Literacy
Production of Product A vs Product B
| Year | Imports | Exports |
|---|---|---|
| 2022 | 80 | 60 |
| 2023 | 90 | 75 |
| 2024 | 100 | 95 |
Compare two or more related series → Multiple Bar Diagram
ఒక total valueను దాని component partsగా subdivide చేసి చూపే bar diagramను Component Bar Diagram అంటారు.
Family Expenditure → Food + Rent + Education + Health
Population → Workers + Non-workers
Budget → Different Expenditure Components
Data percentagesలో ఇచ్చినప్పుడు each bar total heightను 100 unitsగా తీసుకొని componentsను proportionately divide చేయవచ్చు.
| Category | Percentage |
|---|---|
| Food | 40% |
| Rent | 25% |
| Education | 20% |
| Others | 15% |
| Total | 100% |
Every complete bar = 100%
A Pie Diagram is a circle divided proportionately into sectors according to different components of the total.
Pie diagram కూడా component diagram. Difference ఏమిటంటే totalను rectangular barలో కాకుండా circle sectorsగా divide చేస్తారు.
Percentage directly ఇచ్చినప్పుడు:
1% = 3.6°
10% = 36°
25% = 90°
50% = 180°
75% = 270°
100% = 360°
Food = ₹4,000
Rent = ₹3,000
Education = ₹2,000
Others = ₹1,000
Find the sector angle for Food.
Total = 4000 + 3000 + 2000 + 1000 = 10,000
Percentage = Angle ÷ 3.6
| Percentage | Angle |
|---|---|
| 5% | 18° |
| 10% | 36° |
| 12.5% | 45° |
| 20% | 72° |
| 25% | 90° |
| 30% | 108° |
| 40% | 144° |
| 50% | 180° |
| 75% | 270° |
| 100% | 360° |
Grouped frequency distributionsను graphically represent చేయడానికి ఉపయోగించే diagramsను Frequency Diagrams అంటారు.
1. Histogram
2. Frequency Polygon
3. Frequency Curve
4. Ogive
Histogram అనేది continuous frequency distributionను represent చేసే two-dimensional diagram.
Class intervalsను X-axisపై మరియు corresponding frequenciesను Y-axisపై represent చేస్తారు.
X-axis → Class Intervals
Y-axis → Frequencies / Frequency Density
Continuous variableలో one class యొక్క upper boundary next class యొక్క lower boundaryతో coincide అవుతుంది. అందువల్ల rectangles adjacentగా ఉంటాయి.
| Bar Diagram | Histogram |
|---|---|
| Bars have gaps | Rectangles have no gaps |
| Width is generally arbitrary/unimportant for comparison | Width has statistical meaning |
| Height represents magnitude | Area represents frequency |
| Can represent categorical/discrete and other data | Used for continuous frequency distribution |
| Bars need not represent continuous classes | Rectangles correspond to class intervals |
GAPS → Bar Diagram
NO GAPS → Histogram
Class widths equalగా ఉంటే frequenciesను directly rectangle heightsగా use చేయవచ్చు.
| Class | Frequency |
|---|---|
| 0–10 | 5 |
| 10–20 | 8 |
| 20–30 | 12 |
| 30–40 | 7 |
All class widths = 10. Therefore direct frequencies can be compared through heights.
Class intervals unequalగా ఉంటే frequenciesను directly heightsగా తీసుకోవడం misleading కావచ్చు. In such cases Frequency Density ఉపయోగిస్తారు.
Unequal Class Widths → Do not compare raw frequencies directly as heights.
Use → Frequency Density
0–10 → Frequency 20
10–30 → Frequency 30
Find frequency densities.
First class width = 10
Second class width = 20
Unequal intervals → Think Frequency Density immediately.
Histogram can also be used to determine the Mode graphically.
Histogram → Mode
Ogives → Median
A Frequency Polygon is obtained by plotting frequencies against the corresponding class midpoints/class marks and joining successive points by straight lines.
Y-axis → Frequencies
Points → Joined by straight lines
0–10, 10–20, 20–30, 30–40
| Class | Class Mark |
|---|---|
| 0–10 | 5 |
| 10–20 | 15 |
| 20–30 | 25 |
| 30–40 | 35 |
| Frequency Polygon | Histogram |
|---|---|
| Uses plotted points and straight lines | Uses rectangles |
| X-values generally class midpoints | X-axis shows class boundaries/intervals |
| Useful for comparing distributions | Shows shape of continuous distribution |
| Can be derived from histogram | Based on continuous class intervals |
Frequency polygon pointsను closely follow చేస్తూ smooth freehand curve draw చేస్తే దానిని Frequency Curve అంటారు.
Frequency Polygon → Straight Lines
Frequency Curve → Smooth Curve
Smooth curve ప్రతి frequency polygon pointను exactly pass చేయాల్సిన అవసరం లేదు; pointsకు as closely as possibleగా pass అవుతుంది.
Cumulative frequenciesను graphically represent చేసే curveను Ogive అంటారు.
Two types:
2. More-Than Ogive
Less-than cumulative frequenciesను corresponding upper class limits/boundariesకు against plot చేస్తారు.
Y-axis → Less-than Cumulative Frequencies
Less-than ogive generally rises and is never decreasing.
More-than cumulative frequenciesను corresponding lower class limits/boundariesకు against plot చేస్తారు.
Y-axis → More-than Cumulative Frequencies
More-than ogive generally declines and is never increasing.
| Less-Than Ogive | More-Than Ogive |
|---|---|
| Uses less-than cumulative frequency | Uses more-than cumulative frequency |
| Plotted against upper limits | Plotted against lower limits |
| Rising / never decreasing | Declining / never increasing |
Less-than ogive మరియు more-than ogive రెండింటినీ same graphపై draw చేసినప్పుడు వాటి intersection point నుండి medianను graphically determine చేయవచ్చు.
Ogive Intersection → Median
Histogram → Mode
| Class | Frequency | Less-Than C.F. |
|---|---|---|
| 0–10 | 5 | 5 |
| 10–20 | 8 | 13 |
| 20–30 | 12 | 25 |
| 30–40 | 10 | 35 |
| 40–50 | 5 | 40 |
Less-than ogive points:
(20, 13)
(30, 25)
(40, 35)
(50, 40)
For the same data, N = 40.
| Lower Limit | More-Than C.F. |
|---|---|
| 0 | 40 |
| 10 | 35 |
| 20 | 27 |
| 30 | 15 |
| 40 | 5 |
| 50 | 0 |
Less-than C.F. → starts small, ends at N
More-than C.F. → starts at N, ends at 0
Arithmetic Line Graphను Time Series Graph అని కూడా అంటారు.
Variable values over different periods of timeను plot చేసి successive pointsను straight linesతో join చేస్తారు.
Y-axis → Value of Variable
| Year | Production |
|---|---|
| 2021 | 100 |
| 2022 | 120 |
| 2023 | 110 |
| 2024 | 150 |
| 2025 | 180 |
Data changing over Time → Arithmetic Line / Time Series Graph
| Purpose / Data | Suitable Diagram |
|---|---|
| Compare one series across categories | Simple Bar |
| Compare two or more related series | Multiple Bar |
| Show components of a total | Component Bar |
| Show parts of total in a circle | Pie Diagram |
| Continuous frequency distribution | Histogram |
| Frequency vs class midpoint | Frequency Polygon |
| Smooth representation of frequency distribution | Frequency Curve |
| Cumulative frequencies | Ogive |
| Changes over time | Arithmetic Line Graph |
| Problem | Shortcut |
|---|---|
| Pie sector from value | (Value / Total) × 360° |
| Pie sector from percentage | Percentage × 3.6° |
| Percentage from pie angle | Angle ÷ 3.6 |
| Frequency density | f / Class Width |
| Frequency polygon X-value | Class Mark = (L + U)/2 |
| Less-than ogive | Upper Limits + Less-than C.F. |
| More-than ogive | Lower Limits + More-than C.F. |
| Graphical Median | Intersection of two ogives |
| Graphical Mode | Histogram |
Bar diagram has no gaps → FALSE
Ordinary bar diagram has gaps.
Trap 2:
Histogram has gaps → FALSE
Histogram rectangles are adjacent.
Trap 3:
Histogram is mainly for categorical data → FALSE
It represents continuous frequency distribution.
Trap 4:
Pie chart total = 100° → FALSE
Complete circle = 360° = 100%.
Trap 5:
1% in pie chart = 3.6° → TRUE.
Trap 6:
Less-than ogive uses lower limits → FALSE
It uses upper limits/boundaries.
Trap 7:
More-than ogive uses upper limits → FALSE
It uses lower limits/boundaries.
Trap 8:
Frequency polygon uses class midpoints → TRUE.
Trap 9:
Frequency polygon is a smooth curve → FALSE
It uses straight lines.
Trap 10:
Frequency curve must pass exactly through every polygon point → FALSE.
Trap 11:
Intersection of two ogives gives Mode → FALSE
It gives Median.
Trap 12:
Unequal class intervals can always use raw frequencies as histogram heights → FALSE
Frequency density/appropriate adjustment is required.
A) Ogive
B) Frequency curve
C) Pie diagram
D) Frequency polygon
A) Multiple Bar Diagram
B) Histogram
C) Ogive
D) Frequency Curve
A) Component Bar Diagram
B) Frequency Polygon
C) Histogram only
D) Ogive
A) 90°
B) 180°
C) 270°
D) 360°
A) 36°
B) 54°
C) 72°
D) 90°
A) 20%
B) 25%
C) 30%
D) 40%
A) Gaps must exist between rectangles
B) Rectangles are adjacent
C) Only class midpoints are plotted
D) Cumulative frequencies are plotted
A) Continuous frequency distribution
B) Geographical classification only
C) Qualitative attributes only
D) Primary data only
A) Class midpoint
B) Cumulative frequency
C) Frequency density
D) Percentage only
A) Frequency × Class Width
B) Frequency / Class Width
C) Class Width / Frequency
D) Frequency + Class Width
A) Frequencies against class midpoints
B) Cumulative frequencies against upper limits
C) Percentages against years only
D) Frequencies against lower limits only
A) Ogive
B) Pie chart
C) Frequency curve
D) Component bar
A) Histogram
B) Ogive
C) Pie diagram
D) Bar diagram
A) Lower class limits
B) Upper class limits
C) Class midpoints only
D) Frequencies
A) Lower class limits
B) Upper class limits
C) Class marks only
D) Frequencies
A) Mean
B) Mode
C) Median
D) Standard deviation
A) Histogram
B) Pie chart
C) Arithmetic line graph
D) Simple bar only
A) Pie chart
B) Time series graph
C) Frequency curve
D) Component diagram
A) Both A and R are true and R correctly explains A.
B) Both A and R are true but R does not correctly explain A.
C) A is true but R is false.
D) A is false but R is true.
Reason: Histogram is used for continuous frequency distributions.
Reason: The complete circle represents 360°.
Reason: More-than ogive uses lower class limits.
Reason: Histogram areas should appropriately represent frequencies.
| Concept | Must Remember |
|---|---|
| Simple Bar | One series |
| Multiple Bar | Compare two or more series |
| Component Bar | Parts of total |
| Pie Diagram | Circle divided into sectors |
| Complete Pie | 360° = 100% |
| 1% in Pie | 3.6° |
| Histogram | Adjacent rectangles; continuous distribution |
| Unequal Classes | Use Frequency Density |
| Frequency Density | f / Class Width |
| Frequency Polygon | Frequency against Class Midpoint |
| Frequency Curve | Smooth curve |
| Ogive | Cumulative Frequency Curve |
| Less-Than Ogive | Upper Limits |
| More-Than Ogive | Lower Limits |
| Two Ogives Intersect | Median |
| Histogram Graphically Gives | Mode |
| Arithmetic Line Graph | Time Series Graph |
→ Bar + Pie
Bar Diagram
→ Equal-width separated bars
Simple Bar
→ One set of values
Multiple Bar
→ Compare two or more sets
Component Bar
→ Parts of a total
Pie Chart
→ 360° = 100%
Pie Angle
→ (Component / Total) × 360°
Percentage to Angle
→ Percentage × 3.6°
Histogram
→ Continuous distribution + No gaps
Unequal Class Width
→ Frequency Density = f / Width
Frequency Polygon
→ Class Midpoints + Frequencies + Straight Lines
Frequency Curve
→ Smooth Curve
Ogive
→ Cumulative Frequency Curve
Less Than
→ Upper Limits + Rising Curve
More Than
→ Lower Limits + Declining Curve
Intersection of Ogives
→ Median
Histogram
→ Mode graphically
Arithmetic Line Graph
→ Time Series Graph
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