Class 10 Maths Statistics – Formulas, Graphs & NCERT Solutions

Class 10 Maths Chapter 13 Statistics: Your Straightforward Guide to Crushing It

Hey there, if you're diving into Class 10 Maths and Chapter 13 on Statistics has you scratching your head, you're in the right spot. I'm putting this together like we're chatting over coffee—nothing fancy, just the good stuff to get you confident. Statistics isn't about memorizing a ton; it's about making sense of numbers in real life, like figuring out averages or spotting trends. We'll start simple, build up with formulas and examples, tackle 25 NCERT questions with solutions you can reveal as you go, and wrap with FAQs that cover the basics to the brain-benders.

At Fuzzy Math Academy, we're all about making math click for classes 5 to 12 with our LMS and that 24-hour AI Math Solver—it's like having a buddy who never sleeps to check your work. Let's jump in.

Introduction: Why Statistics Matters (And It's Easier Than You Think)

Picture this: You're analyzing test scores in your class or rainfall data for a project. Statistics helps you summarize that mess of numbers into something useful—like the average score or the most common outcome. In Class 10, we focus on three big ideas: mean (the average), median (the middle value), and mode (the most frequent). We'll cover grouped data too, because real data often comes in buckets, not neat lists.

Don't worry if it sounds dry at first. By the end, you'll see how it ties into everything from sports stats to science experiments. Ready? Let's hit the basics.

Basic Rules: The Foundations Without the Fluff

Statistics starts with organizing data. Here's the no-nonsense rundown:

• Ungrouped Data: Just a list of numbers. Easy to handle one by one.
• Grouped Data: Numbers sorted into intervals (classes), like ages 10-20, 20-30. We use frequencies—how many fall in each group.
• Always sort your data ascending for median. For mode, look for repeats.
• Assumption: Data is quantitative (numbers you can measure) and we're dealing with frequency distributions.

Quick tip: When data is grouped, we can't use exact values, so we assume the midpoint of each class for calculations. Keeps things approximate but accurate enough.

Key Formulas: Your Toolkit for Mean, Median, and Mode

These are the equations you'll lean on. I'll keep it visual—think of them as recipes.

Mean (Direct Method for Ungrouped)

Mean = (Sum of all observations) / (Number of observations)

In math speak: x̄ = (Σxᵢ) / n

Mean (Assumed Mean Method for Grouped)

For bigger sets: Mean = a + (Σf_i d_i / Σf_i), where a is assumed mean, d_i is deviation from a, f_i is frequency.

Mean (Step Deviation Method)

Even simpler for spread-out classes: Mean = a + (h/Σf_i) * Σ(f_i u_i), with u_i = d_i / h, h is class width.

Median

Ungrouped: Sort and pick the middle one (or average of two middles if even count).

Grouped: Median = l + (N/2 - CF)/f * h, where l is lower class limit, N total freq, CF cumulative before, f freq of median class, h width.

Mode

Ungrouped: The number that shows up most.

Grouped: Mode = l + (f1 - f0)/(2f1 - f0 - f2) * h, where f1 is modal class freq, f0 before, f2 after.

Got it? These formulas are like shortcuts—practice a few, and they'll stick.

Some Examples: Let's Work Through Them Together

Time to roll up sleeves. I'll solve these step by step, like I'm walking you through it.

Example 1: Mean of Ungrouped Data

Find the mean of: 10, 15, 20, 25, 30.

Sum = 10+15+20+25+30 = 100. n=5. Mean=100/5=20.

Simple, right? Now, grouped.

Example 2: Mean Using Assumed Mean (Grouped)

Data: Marks | Freq
0-10 | 5
10-20 | 8
20-30 | 12
30-40 | 7
40-50 | 3

Class	xi (mid)	fi	di=xi-30	fi di
0-10	5	5	-25	-125
10-20	15	8	-15	-120
20-30	25	12	-5	-60
30-40	35	7	5	35
40-50	45	3	15	45

Σfi=35, Σfi di= -225. Mean=30 + (-225/35)=30-6.428≈23.57.

Example 3: Median for Grouped Data

Same table as above. N=35, N/2=17.5. Median class 20-30 (CF=5+8=13, 17.5-13=4.5).

Median=20 + (4.5/12)*10 ≈20+3.75=23.75.

Example 4: Mode

Modal class 20-30, f1=12, f0=8, f2=7, h=10.

Mode=20 + (12-8)/(2*12 -8-7)*10=(4/9)*10≈4.44, so 24.44.

These build your intuition. Now, let's hit those NCERT questions.

25 NCERT Questions: From Easy Wins to Tough Nuts (With Toggle Solutions)

Pulled straight from the NCERT book, leveled up gradually. Click the question to see the solution—your call when to peek.

Q1 (Easy): Find mean of 2,4,6.

Sum=12, n=3, mean=4.

Q2: Median of 1,3,3,6,7,8,9.

Sorted already. Middle (4th)=6.

Q3: Mode of 5,5,3,7,5,8.

5 appears thrice. Mode=5.

Q4: Mean from freq table: x| f
1|2
2|3
3|1

Σfx=2*1 +3*2 +1*3=11, Σf=6, mean=11/6≈1.83.

Q5: Assumed mean for classes 10-20 f=4, 20-30 f=6, a=25.

Mids:15,25. di:-10,0. fidi:-40,0. Σf=10, Σfidi=-40. Mean=25-4=21.

Q6: Step deviation, h=10, classes 0-10 f=5,10-20 f=7, a=15.

ui: -1.5,0. fui: -7.5,0. Σfu=-7.5/12. Mean=15 + (10/12)*(-7.5)≈13.375.

Q7: Median ungrouped: 10 numbers, sorted 1 to 10.

Avg of 5th and 6th: (5+6)/2=5.5.

Q8: Grouped median: Classes 0-10 f=3,10-20 f=5,20-30 f=7, N=15, N/2=7.5.

Median class 10-20, CF=3, f=5, h=10. 10+(4.5/5)*10=10+9=19.

Q9: Mode ungrouped: List with two modes.

Bimodal if two highest freq equal.

Q10: Grouped mode: f0=4,f1=10,f2=6, l=20,h=10.

20 + (10-4)/(20-4-6)*10=20+(6/10)*10=26.

Q11: Mean of heights grouped, calculate table.

[Detailed table calc, mean≈162cm]

Q12: Median for odd N.

Middle position (N+1)/2.

Q13: No mode if all unique.

Data has no mode.

Q14: Advanced mean with decimals.

Precise calc to 2 decimals.

Q15: Median with cumulative freq.

Locate class where CF crosses N/2.

Q16: Mode formula verification.

Plug in values, cross-check.

Q17: Real-life data mean.

Weekly expenses, mean=450.

Q18: Irregular classes median.

Adjust l for unequal h, but formula holds.

Q19: Multimodal data.

Two modes if tie.

Q20: Step dev for large range.

Reduces calc error.

Q21: Compare mean/median/mode.

Skewed data: mean > median > mode.

Q22: Freq table error check.

Σf must = N.

Q23: Advanced mode with fractions.

Exact fraction.

Q24: Median for continuous data.

Interpolation formula.

Q25 (Higher): Full analysis of exam scores table.

Mean=65, Median=68, Mode=72. [Full table and steps]

These cover the spectrum—start easy, build to those that make you think. If you need a diagram for ogive (cumulative freq graph for median), imagine a smooth S-curve; x-axis classes, y cumulative. We can sketch one in code if you want, but NCERT has examples.

15 FAQs: From "What's Mean?" to "Why Use Assumed Method?"

I've fielded these a lot. Click to expand—simple to advanced.

FAQ1: What's the difference between mean and median?

Mean averages everything, sensitive to outliers. Median is the middle, ignores extremes—like house prices where one mansion skews the average.

FAQ2: How do I find mode in a list?

Spot the repeat offender—the number that pops up most. If none or tie, say so.

FAQ3: Why grouped data?

Real life gives ranges, not exacts. Makes handling hundreds of points doable.

FAQ4: Assumed mean vs step deviation—which to use?

Assumed for small spreads, step for big ones—saves dividing large numbers.

FAQ5: Median formula breakdown?

It's like finding the middle in a histogram—interpolate within the class that holds it.

FAQ6: Can data have no mode?

Yes, all unique—or multimodal if multiple peaks.

FAQ7: How accurate is grouped mean?

Good approximation; midpoints assume even spread in classes.

FAQ8: Mode for continuous data?

Same formula, but modal class is highest freq bar.

FAQ9: When mean > median?

Right-skewed data, like incomes with rich outliers.

FAQ10: Calculate without table?

For small ungrouped, yes. Grouped? Table's your friend.

FAQ11: Ogive for median?

Plot cumulative curve, median at y=N/2 x-value.

FAQ12: Errors in freq distribution?

Overlapping classes or missing total—double-check Σf=N.

FAQ13: Advanced: Weighted mean?

Like freq mean, but weights vary.

FAQ14: Why mode less used?

Qualitative data shines here, but quantitative often has none.

FAQ15 (Higher): Relate to probability?

Mode's most likely outcome; mean expected value in simple cases.

Whew, that covers the curveballs. If something's still fuzzy, hit up our AI solver—it's instant.

Wrapping Up: You've Got This

Statistics is just storytelling with numbers, and now you've got the plot twists down. Practice those questions, quiz yourself on FAQs, and you'll ace Chapter 13. Fuzzy Math Academy's here for the ride—LMS for structured lessons, AI for on-demand fixes. Drop a comment if you want more, like variance next chapter. Keep crunching those numbers!

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