Overview
Chapter: Data Handling (Mathematics – VIII) introduces students to collecting, organising, representing and interpreting numerical information. It explains types of data, how to prepare frequency distributions (grouped and ungrouped), and how to display data visually using pictographs, bar graphs, histograms, frequency polygons and ogives. The chapter also develops measures of central tendency — mean, median and mode — (for ungrouped and grouped data), and basic measures of spread such as range. Emphasis is on real-life applications (surveys, experiments, classroom data), reading and drawing graphs accurately, and using these tools to make simple inferences and informed decisions. This chapter builds foundations for further study in statistics and probability.
Learning Objectives
- Define terms such as data, raw data, frequency, class interval and frequency distribution, and identify them in examples
- Collect and organize raw data into frequency tables for discrete and grouped (continuous) data
- Construct and interpret bar graphs and double bar graphs from given frequency distributions
- Draw and interpret histograms and frequency polygons for grouped data
- Represent data using pie charts and calculate central angles from frequencies
- Calculate mean, median and mode for ungrouped (discrete) data and interpret the results
- Estimate mean, median and mode for grouped (continuous) data using class mid-points and apply them to solve problems
- Compute the combined mean of two or more data sets and apply it to contextual problems
Topics in this chapter
6 topics · tap a topic title to jump straight to it.
Introduction to Data
Introduction to Data
Key Point: Mean (ungrouped/raw data): \u03BC or x̄ = (x1 + x2 + ... + xn) / n
What is data? Data are facts or numbers collected for analysis. In mathematics and statistics, data help us describe, compare and make decisions. Examples: marks of students, daily rainfall, favourite colours in a class.
Types of data
- Qualitative (categorical): non-numeric labels (e.g., blood group, favourite sport).
- Quantitative (numerical): numeric values. These are of two kinds:
- Discrete: whole numbers (e.g., number of siblings, test scores).
- Continuous: can take any value in a range (e.g., height, weight, time).
- Primary data: collected directly (surveys, experiments).
- Secondary data: obtained from other sources (books, internet).
From raw data to information — key steps
- Collection — decide what to measure and how (sample, questionnaire).
- Organization — list raw data, use tally marks and make a frequency table or grouped frequency distribution.
- Representation — draw graphs (pictograph, bar graph, histogram, pie chart, frequency polygon) to visualise patterns.
- Interpretation — compute summary measures (mean, median, mode, range) and draw conclusions.
Frequency table and tally marks — Raw observations are grouped into classes or categories. Use tally marks (||||) to count data quickly, then convert tally counts to frequencies.
Summary measures (intuitively) — mean gives the average, median gives the middle value, mode gives the most frequent value, range gives spread between largest and smallest values.
Good practices: label axes, choose an appropriate scale, use equal class width for histograms, include title and units, avoid misleading visuals (distorted scales).
- Example 1 — Marks of 7 students: 48, 55, 62, 55, 70, 48, 62. Frequency table: 48(2),55(2),62(2),70(1). Mean = (48+55+62+55+70+48+62)/7 = 400/7 ≈57.14. Median = middle value when sorted (48,48,55,55,62,62,70) → 55. Mode = 48,55 and 62 (all occur twice) → multimodal. Range = 70−48 = 22.
- Example 2 — Survey for favourite fruit in class (categories): Apple(8), Banana(5), Mango(10), Orange(7). Use a pictograph: choose 1 symbol = 1 student and draw symbols or use scale (1 symbol = 2 students).
- Example 3 — Heights of students grouped (cm): 140–144(3),145–149(6),150–154(8),155–159(3). Draw a histogram with class intervals on x-axis and frequency on y-axis; ensure equal class width and contiguous bars.
- Example 4 — Number of books read in a month by 10 students: 0,1,2,0,3,2,1,4,2,1. Frequency: 0(2),1(3),2(3),3(1),4(1). Mode = 1 and 2 (both most frequent); Mean = (0+1+2+0+3+2+1+4+2+1)/10 = 16/10 = 1.6; Median = average of 5th and 6th sorted values → (1+2)/2 = 1.5.
- \[Mean (ungrouped/raw data): \u03BC or x̄ = (x1 + x2 + ... + xn) / n\]
- \[Mean (frequency data): x̄ = (\u03Sigma f_i x_i) / N where f_i is frequency and N = \u03Sigma f_i\]
- \[Median (ungrouped): Sort data\]\[If n is odd → middle value at position (n+1)/2\]\[If n is even → median = average of values at positions n/2 and n/2 + 1.\]
- \[Mode: The value(s) with highest frequency. (For grouped data use the modal class\]\[advanced formula for grouped data: Mode ≈ L + [(f1 - f0) / (2f1 - f0 - f2)] * h where L = lower boundary of modal class\]\[f1 = frequency of modal class\]\[f0 = frequency of previous class\]\[f2 = frequency of next class\]\[h = class width.)\]
- \[Range: Range = Maximum value − Minimum value\]
- \[Relative frequency: rf_i = f_i / N\]
Collection and Recording of Data
Collection and Recording of Data
Key Point: Total frequency (N) = Σ f_i (sum of all class frequencies)
What is data? Data are facts or measurements collected for analysis. In mathematics (data handling) we collect numbers or categories about objects, people or events to describe, compare or draw conclusions.
Types of data
- Qualitative (categorical): names or categories (e.g., favourite colour, blood group).
- Quantitative (numerical): numbers (e.g., heights, ages). Quantitative data can be discrete (countable values like number of books) or continuous (measurable values like height, temperature).
Sources of data
- Primary data — collected first-hand by survey, experiment or observation (e.g., asking classmates their favourite fruit).
- Secondary data — obtained from books, articles, government reports or the internet (e.g., population data in a report).
Steps in collection
- Define the objective (what you want to find).
- Decide what data to collect (which variables and their types).
- Choose method: survey/questionnaire, interview, measurement or observation.
- Design a simple, clear questionnaire or measurement plan. Use precise questions and consistent units.
- Collect data carefully and record immediately to avoid errors.
Recording data
Raw data are organised into tables for clarity. For surveys and counts use tally marks to keep a quick record and then convert to a frequency table:
- Tally marks: group marks in fours and cross the fifth (|||| = 5). Useful while collecting.
- Frequency table: list each distinct value or class interval and its frequency (number of occurrences).
Grouped data (for continuous or many different values) are recorded using class intervals. Important terms:
- Class interval: range like 130–139 cm.
- Class width: upper boundary minus lower boundary (or next lower bound)
- Class mark (midpoint): (lower limit + upper limit)/2.
- Cumulative frequency: running total of frequencies up to a class.
Why organise data? Organised data (tables and charts) makes it easier to see patterns, compare groups and prepare graphs like bar graphs, histograms, pie charts or ogives.
Good practices: use consistent units, include all observations, check for errors, label columns and headings, and choose suitable class sizes so the table is neither too detailed nor too coarse.
- Survey in class: Ask 40 students their favourite sport. Record answers with tally marks, then produce a frequency table (Cricket: |||| ||, Football: |||| |, Badminton: ||||, etc.). Use a bar graph to show the results.
- Measuring heights: Measure 30 students' heights (in cm). Group them into class intervals 140–149, 150–159, 160–169, etc., count frequencies, compute class marks and cumulative frequencies for further analysis.
- Daily temperatures: Record maximum temperature each day for a month (primary data). Convert raw numbers to a frequency distribution with suitable class intervals and draw a histogram to visualise temperature distribution.
- Counting books: Record the number of books read by 25 students in a year (discrete data). Make a frequency table and use a bar graph or pie chart to compare counts by categories (0–2, 3–5, 6+).
- \[Total frequency (N) = Σ f_i (sum of all class frequencies)\]
- \[Class mark (midpoint) m_i = (lower limit + upper limit) / 2\]
- \[Class width = upper limit − lower limit (or next class lower bound − this class lower bound)\]
- \[Cumulative frequency for class k = f_1 + f_2 + ... + f_k\]
- \[Relative frequency of class i = f_i / N\]
- \[Percentage frequency = (f_i / N) × 100\]
Frequency Distribution and Tables
Frequency Distribution and Tables
Key Point: Frequency (f): count of observations in a class (no formula — it's a count).
What is a frequency distribution? A frequency distribution organises raw data into classes or categories and shows how often (frequency) each value or class occurs. It makes large data sets easier to understand and analyse.
Types: 1) Ungrouped (discrete) frequency table — used when data take a small number of distinct values (e.g., shoe sizes). 2) Grouped frequency table — used for continuous or large-range data; data are put into class intervals (e.g., ages, marks).
Key components and terms: Class interval (e.g., 10–14), Class limits (lower and upper limits), Class boundaries (adjusted for continuous data), Frequency (f) — number of observations in a class, Cumulative frequency (CF) — running total up to a class, Relative frequency — fraction or proportion of total observations in a class, Class mark (midpoint) — (lower + upper)/2.
Steps to prepare a grouped frequency distribution: 1) Find the range = max − min. 2) Decide number of classes (k) — commonly 5–15 depending on data size. 3) Calculate class width ≈ (range)/k and round up to a convenient value. 4) Choose a suitable starting point and form k equal-width classes. 5) Tally the data into classes and count frequencies. 6) Compute cumulative and relative frequencies if required.
Simple example table (grouped):
| Class | Frequency (f) |
|---|---|
| 12–14 | 4 |
| 15–17 | 5 |
| 18–20 | 3 |
| 21–23 | 2 |
| Total | 14 |
This table shows how many observations fall in each interval, making patterns easy to spot.
Why use frequency tables? They summarise data compactly, reveal distribution shape (e.g., skewed or symmetric), allow calculation of medians, modes and drawing graphs (histogram, ogive, frequency polygon) used for comparison and interpretation.
- Marks of 30 students: create a grouped table (e.g., 0–10, 11–20, 21–30, ...) and count how many students fall in each class to see performance distribution.
- Daily high temperatures for a month: group temperatures (e.g., 20–22°C, 23–25°C, ...) to observe common temperature ranges.
- Ages of participants in a workshop: for many ages use class intervals (e.g., 10–19, 20–29, 30–39) to understand age-groups present.
- Shoe sizes recorded in a store: use an ungrouped frequency table listing each size and its frequency, then find the most common shoe size (mode).
- \[Frequency (f): count of observations in a class (no formula — it's a count).\]
- \[Total number of observations (N): N = Σ f_i.\]
- \[Relative frequency: r_i = f_i / N.\]
- \[Percentage frequency: p_i = (f_i / N) × 100%.\]
- \[Cumulative frequency (CF) for class k: CF_k = Σ_{i=1 to k} f_i.\]
- \[Class width (approx.): width ≈ (max − min) / k (round up to a convenient number).\]
Graphical Representation of Data
Graphical Representation of Data
Key Point: Class width = (Upper boundary − Lower boundary) of a class
What it is
Graphical representation of data means displaying data visually — using graphs and charts — so patterns, comparisons and trends are easy to see. It converts numerical or categorical data into pictorial form.
Types of data
- Qualitative (categorical) — e.g., favourite sport, blood group.
- Quantitative — numerical: discrete (countable values, e.g., number of students) or continuous (measured values, e.g., marks, weight).
Common graphical forms and when to use them
- Bar graph (vertical or horizontal) — for comparing counts/values of categories (discrete or grouped). Bars have equal width and gaps between them.
- Histogram — for continuous data grouped into class intervals; adjacent bars touch (no gaps) since intervals are continuous.
- Pie chart — for showing proportions of a whole (percentages or fractions).
- Line graph — for showing trends over time.
- Frequency polygon — join midpoints of top of histogram bars with straight lines; useful for comparing distributions.
- Ogive (cumulative frequency graph) — plots cumulative frequency vs class boundary to read medians, percentiles.
Steps to draw most graphs
- Prepare a frequency distribution table (for grouped data, list class intervals and frequencies).
- Decide the type of graph appropriate to the data.
- Choose scales for the axes so that the data fits neatly (use simple, regular intervals).
- Label axes clearly (variable name and units) and give a meaningful title.
- Draw the graph: bars for bar graph, contiguous bars for histogram, measure angles for pie chart, plot points and join for line graph or ogive.
- Add legend or colour-coding when multiple data sets are shown.
Important points and tips
- For a histogram ensure class intervals are of equal width for simple frequency comparison; if unequal, use frequency density (frequency ÷ class width) on the vertical axis.
- For pie chart, calculate each sector angle = (frequency/total) × 360°.
- For frequency polygon use class midpoints on the x-axis and join the plotted points; close the polygon at both ends by dropping to zero at one class width beyond the first and last midpoints.
- Ogive plots cumulative frequency against upper class boundaries (or lower boundaries depending on convention) and is useful to obtain median and quartiles graphically.
- Marks of 40 students grouped into class intervals: 0–9, 10–19, …, 90–99. Make a histogram: compute frequencies per interval, choose appropriate scale, draw contiguous bars whose heights equal the frequencies.
- Survey of favourite fruit among 100 students: Apple 30, Banana 20, Mango 25, Orange 15, Others 10. Make a pie chart: compute angles: Apple = (30/100)×360 = 108°, Banana = 72°, Mango = 90°, Orange = 54°, Others = 36° and draw sectors accordingly.
- Monthly rainfall (mm) for a year: plot a line graph with months on x-axis and rainfall on y-axis to show trend and identify rainy months and dry months.
- A grouped frequency table with unequal class widths: use frequency density (frequency ÷ class width) on the vertical axis when drawing the histogram so areas represent frequencies correctly.
- Construct an ogive from class intervals and frequencies by computing cumulative frequency and plotting cumulative frequency against the upper class boundaries; use it to estimate the median where cumulative frequency = N/2.
- \[Class width = (Upper boundary − Lower boundary) of a class\]
- \[Class midpoint = (Lower limit + Upper limit) / 2\]
- \[Relative frequency = frequency / total frequency\]
- \[Percentage frequency = (frequency / total frequency) × 100\]
- \[Angle for pie chart sector (in degrees) = (frequency / total frequency) × 360\]
- \[Frequency density (for unequal class widths) = frequency / class width\]
Measures of Central Tendency
Measures of Central Tendency
Key Point: Mean (ungrouped) = (Sum of observations) / (Number of observations) = (Σx)/n
What are Measures of Central Tendency?
Measures of central tendency are numbers that describe the center or typical value of a data set. The three main measures are mean (average), median (middle value), and mode (most frequent value). They help summarize large data sets with a single representative value.
When to use which?
- Mean: Useful for numerical data without extreme outliers and when every value matters (e.g., average marks).
- Median: Better when data has outliers or is skewed (e.g., income data). It is the middle value.
- Mode: Useful for categorical or discrete data to find the most common category or value (e.g., most common shoe size).
How to find them (basic steps)
- Mean (ungrouped): Add all observations and divide by number of observations.
- Median (ungrouped): Order observations. If n is odd, median is the ((n+1)/2)-th value; if n is even, it is the average of the (n/2)-th and (n/2 + 1)-th values.
- Mode (ungrouped): The value that appears most often. There can be more than one mode.
Grouped data: For class intervals we use class-marks (mid-points) and frequencies. Special formulas are used to compute mean, median and mode for grouped frequency distributions (see formulas below).
Interpretation and caution: Mean uses all values so it is sensitive to extreme values. Median resists extremes. Mode gives the most typical category but may not reflect central tendency well if distribution is flat or has several peaks.
- Mean (ungrouped): Marks scored by 5 students: 62, 75, 80, 90, 83. Mean = (62+75+80+90+83)/5 = 390/5 = 78.
- Median (ungrouped): Heights (cm) of 6 students: 140, 142, 145, 147, 150, 152. Ordered list is same; n=6 even so median = average of 3rd and 4th values = (145+147)/2 = 146.
- Mode (ungrouped): Shoe sizes of students: 6, 7, 6, 8, 6, 7. Mode = 6 (appears most often).
- Mean (grouped): Class intervals 100-119 (2), 120-139 (5), 140-159 (8), 160-179 (5) where numbers in parentheses are frequencies. Class-marks: 109.5,129.5,149.5,169.5. Mean = (Σ f*x)/Σ f = (2*109.5 + 5*129.5 + 8*149.5 + 5*169.5)/20 = (219 + 647.5 + 1196 + 847.5)/20 = 290.0/20 = 145.0 (approx).
- Median (grouped): Using same grouped data above, n = 20, n/2 = 10. Find median class where cumulative frequency crosses 10 (the 10th observation). Apply median formula for grouped data (see formulas).
- Mode (grouped): For grouped frequency, mode is estimated from the modal class (class with highest frequency) using the grouped mode formula (see formulas).
- \[Mean (ungrouped) = (Sum of observations) / (Number of observations) = (Σx)/n\]
- \[Mean (grouped) = (Σ f * x) / Σf\]\[where x = class mark (mid-point) and f = frequency of the class\]
- \[Assumed mean method (grouped): Mean = A + h*(Σ f*d / Σf)\]\[where A = assumed mean (a class mark)\]\[d = (x - A)/h\]\[h = class width\]
- \[Median (ungrouped): If n odd: median = value at position (n+1)/2\]\[If n even: median = average of values at positions n/2 and n/2 + 1\]
- \[Median (grouped): Median = L + [(n/2 - c.f) / f] * h\]\[where L = lower boundary of median class\]\[c.f = cumulative frequency before median class\]\[f = frequency of median class\]\[h = class width\]
- \[Mode (ungrouped): The value with the highest frequency (may be more than one).\]
Interpretation and Application of Data
Interpretation and Application of Data
Key Point: Ungrouped mean (arithmetic mean): \u003cmean\u003e = (x_1 + x_2 + ... + x_n) / n
What it means: Interpretation of data is reading, understanding and drawing conclusions from collected data (tables, charts, graphs). Application of data is using those conclusions for decision-making (planning, predicting, comparing, solving problems).
Steps to interpret data:
- Identify the type of data and how it is organized (raw list, frequency distribution, grouped classes).
- Look at central tendency (mean, median, mode) to find a typical value.
- Check spread/variation (range, class width, distribution shape) to see how values are spread.
- Observe trends, peaks, gaps, and outliers that affect conclusions.
- Compare datasets using relative measures (percentages, ratios) or visual comparison (double bar charts, side-by-side box plots).
- Translate results into practical actions: estimate, predict or recommend based on evidence.
How to apply data (common uses):
- School: analyze class test scores to find topics needing revision.
- Business: use sales data to stock popular items and reduce slow-moving items.
- Government: use population and age-distribution data to plan schools or hospitals.
- Weather and agriculture: use rainfall data to schedule crops or irrigation.
Important interpretation tips for Class 8:
- Always check total frequency (N) — many percentages and angles depend on it.
- For grouped data, use class marks to compute averages; for precise median and mode there are standard formulas.
- Choose the right graph: bar-chart for categories, histogram for continuous grouped data, pie-chart for part–of–whole, ogive for cumulative comparisons.
- Remember: correlation seen in data does not prove causation — ask why the pattern exists.
- School marks: Given frequencies of marks in intervals 0–10, 11–20, ... find mean marks (grouped mean), identify the modal class and estimate the median to see if most students passed.
- Survey of favourite fruits: 120 students choose between apple, banana, mango and orange. Use a bar graph to compare popularity and a pie chart to show percentage share; take decisions about stocking fruit for a school event.
- Monthly rainfall (in mm) for a town: draw a line graph to spot months with highest/lowest rainfall and an ogive to find how many months have rainfall below a certain value.
- Shop sales by product category: compute percentage share ((category sales / total sales) * 100) and use a pie chart to visualise contribution of each category to total sales.
- \[Ungrouped mean (arithmetic mean): \u003cmean\u003e = (x_1 + x_2 + ... + x_n) / n\]
- \[Grouped mean (using class mark m_i): \u003cmean\u003e = (Σ f_i m_i) / Σ f_i\]\[where f_i is frequency and m_i = (lower limit + upper limit)/2\]
- \[Median for ungrouped data: arrange data in order\]\[median is the ((n+1)/2)-th value if n is odd\]\[or average of (n/2)-th and (n/2 +1)-th values if n is even\]
- \[Median for grouped data: Median = L + [(N/2 - cfb) / f_m] × h\]\[where L = lower boundary of median class\]\[N = total frequency\]\[cfb = cumulative frequency before median class\]\[f_m = frequency of median class\]\[h = class width\]
- \[Mode for ungrouped data: the value with highest frequency\]\[For grouped data (estimated mode): Mode = L + [(f_m - f_1) / (2f_m - f_1 - f_2)] × h\]\[where f_m = frequency of modal class\]\[f_1 and f_2 = frequencies of preceding and succeeding classes\]
- \[Cumulative frequency (CF): CF_k = f_1 + f_2 + ... + f_k\]
Key Concepts
- Data
- Collected values, facts or measurements from observations or experiments.
- Raw Data
- Unprocessed data presented in its original form as a list of observations.
- Observation
- A single measured or recorded value in a data set.
- Frequency
- The number of times a particular observation or value occurs in a data set.
- Frequency Distribution
- An organized representation (often a table) showing values or class intervals and their frequencies.
- Class Interval
- A range of values grouped together in a frequency distribution for continuous data.
- Class Width
- The difference between the upper and lower boundaries or limits of a class interval.
- Tally Marks
- A quick way to record and count frequencies using groups of marks, usually in sets of five.
- Discrete Data
- Data that can take only specific, separate values (often counts).
- Continuous Data
- Data that can take any value within a range (measurements), including decimals.
- Relative Frequency
- The ratio of the frequency of a value or class to the total number of observations (often a fraction or percentage).
- Cumulative Frequency
- The running total of frequencies up to and including a given class or value.
- Mode
- The value(s) that occur most frequently in a data set.
- Median
- The middle value in an ordered data set; if even number of observations, the median is the average of the two middle values.
- Mean (Average)
- The sum of all observations divided by the number of observations.
- Range
- Difference between the maximum and minimum values in a data set.
- Pictograph
- A chart that uses pictures or symbols to represent quantities, with each symbol standing for a fixed number of items.
- Bar Graph
- A chart using separate bars of equal width to represent frequencies of different categories; bars are spaced apart.
- Histogram
- A chart for grouped continuous data where adjacent bars represent class intervals; area or height shows frequency.
- Pie Chart
- A circular chart divided into sectors where each sector's angle (or area) is proportional to the frequency of a category.
Practice Questions
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The most frequently occurring value in a data set is called: (a) Mean (b) Median (c) Mode (d) Range / किसी डेटा सेट में सबसे अधिक बार आने वाले मान को कहते हैं: (a) माध्य (b) माध्यिका (c) बहुलक (d) परिसर
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(c) Mode / बहुलक — The mode is the observation with the highest frequency in a data set. A data set can have more than one mode (multimodal). / बहुलक वह प्रेक्षण है जिसकी आवृत्ति सबसे अधिक है। एक डेटा सेट में एक से अधिक बहुलक भी हो सकते हैं।
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The range of a data set is equal to ________ minus ________. / एक डेटा सेट का परिसर ________ घटा ________ के बराबर होता है।
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Maximum value minus Minimum value / अधिकतम मान घटा न्यूनतम मान — Range = Maximum value − Minimum value. It gives the spread of the data. / परिसर = अधिकतम मान − न्यूनतम मान। यह डेटा के विस्तार को दर्शाता है।
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Which type of graph is most appropriate for displaying grouped continuous data (like heights of students in class intervals)? (a) Pie chart (b) Bar graph (c) Histogram (d) Pictograph / समूहबद्ध सतत डेटा (जैसे वर्ग अंतरालों में छात्रों की ऊँचाइयाँ) प्रदर्शित करने के लिए कौन-सा ग्राफ सबसे उपयुक्त है? (a) पाई चार्ट (b) दंड ग्राफ (c) आयत चित्र (d) चित्रालेख
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(c) Histogram / आयत चित्र — A histogram uses adjacent (touching) bars for class intervals of continuous data, where the height of each bar represents the frequency of that class. / आयत चित्र में सतत डेटा के वर्ग अंतरालों के लिए सटे हुए दंड होते हैं, जहाँ प्रत्येक दंड की ऊँचाई उस वर्ग की आवृत्ति दर्शाती है।
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The marks scored by 5 students are: 40, 50, 60, 70, 80. The mean is ________. / 5 छात्रों के अंक हैं: 40, 50, 60, 70, 80। माध्य ________ है।
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60 — Mean = (40 + 50 + 60 + 70 + 80) / 5 = 300 / 5 = 60. / माध्य = (40 + 50 + 60 + 70 + 80) / 5 = 300 / 5 = 60।
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The median of the data set 3, 7, 8, 12, 15 (already arranged in order) is: (a) 8 (b) 9 (c) 7 (d) 12 / डेटा सेट 3, 7, 8, 12, 15 (पहले से क्रम में) की माध्यिका है: (a) 8 (b) 9 (c) 7 (d) 12
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(a) 8 — For an odd number of observations (n=5), the median is the middle value at position (5+1)/2 = 3rd position, which is 8. / विषम संख्या में प्रेक्षण (n=5) के लिए, माध्यिका मध्य का मान होती है जो (5+1)/2 = तीसरी स्थिति पर है, अर्थात 8।
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True or False: Primary data is data obtained from already published sources like books or the internet. / सत्य या असत्य: प्राथमिक डेटा वह डेटा है जो पहले से प्रकाशित स्रोतों जैसे किताबों या इंटरनेट से प्राप्त किया गया हो।
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False / असत्य — Primary data is collected directly by the researcher through surveys, experiments or observations. Data from published sources is called secondary data. / प्राथमिक डेटा शोधकर्ता द्वारा सर्वेक्षण, प्रयोग या अवलोकन के माध्यम से सीधे एकत्र किया जाता है। प्रकाशित स्रोतों से प्राप्त डेटा को द्वितीयक डेटा कहते हैं।
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To draw a pie chart, the angle for each sector is calculated as: sector angle = (frequency ÷ total frequency) × ________. / पाई चार्ट बनाने के लिए, प्रत्येक सेक्टर का कोण इस प्रकार ज्ञात करते हैं: सेक्टर कोण = (आवृत्ति ÷ कुल आवृत्ति) × ________।
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360° — Since a full circle = 360°, each sector angle is its proportion of 360°. Example: if frequency is 30 out of 120, angle = (30/120) × 360° = 90°. / 360° — चूँकि एक पूर्ण वृत्त = 360°, प्रत्येक सेक्टर का कोण 360° का उसका अनुपात होता है। उदाहरण: यदि आवृत्ति 120 में से 30 है, तो कोण = (30/120) × 360° = 90°।
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Explain the difference between a bar graph and a histogram, giving one key feature of each. / एक दंड ग्राफ और एक आयत चित्र के बीच अंतर स्पष्ट करें, प्रत्येक की एक मुख्य विशेषता बताते हुए।
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Bar graph: Used for categorical or discrete data; bars are separated (gaps between bars) and each bar represents a specific category. Histogram: Used for grouped continuous data; bars are adjacent (no gaps) and each bar represents a class interval. / दंड ग्राफ: श्रेणीबद्ध या असंतत डेटा के लिए; दंडों के बीच अंतराल होते हैं और प्रत्येक दंड एक विशेष श्रेणी को दर्शाता है। आयत चित्र: समूहबद्ध सतत डेटा के लिए; दंड सटे हुए होते हैं (कोई अंतराल नहीं) और प्रत्येक दंड एक वर्ग अंतराल को दर्शाता है।
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