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Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

• Identify the Problem
• Define the Problem
• Form a Strategy
• Organize Information
• Allocate Resources
• Monitor Progress
• Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

• Asking questions about the problem
• Breaking the problem down into smaller pieces
• Looking at the problem from different perspectives
• Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

• Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
• Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

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You can become a better problem solving by:

• Practicing brainstorming and coming up with multiple potential solutions to problems
• Being open-minded and considering all possible options before making a decision
• Breaking down problems into smaller, more manageable pieces
• Asking for help when needed
• Researching different problem-solving techniques and trying out new ones
• Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Math Statistics: Mean

In these lessons, we will learn how to calculate the mean of a given data set. We will also learn how to solve word problems involving mean.

Related Pages Center Of A Distribution Median Mode Central Tendency More Statistics Lessons

The mean of a set of observations is the average. It is obtained by dividing the sum of data by the number of observations.

The formula is:

Example: Find the mean of the following set of integers. 8, 11, –6, 22, –3

Example: The set of scores 12, 5, 7, -8, x, 10 has a mean of 5. Find the value of x.

How to calculate the mean, median, and mode and range of a given data set?

Word Problems Involving Mean

When there are changes in the number or the values of the observations in a set, the mean will be changed.

Example: The mean score of a group of 20 students is 65. Two other students whose scores are 89 and 85 were added to the group. What is the new mean of the group of students?

can rewritten as: Total score = Mean × Number of students Total score of the original group = 65 × 20 = 1,300

Total score of the new group = Total score of the original group + scores of the 2 new students = 1,300 + 89 + 85 = 1,474

Number of students in the new group = Number of students in the original group + Number of new data = 20 + 2 = 22

Example: The mean of a list of 6 numbers is 20. If we remove one of the numbers, the mean of the remaining numbers is 15. What is the number that was removed?

Solution: Using the formula: Sum = Mean × Number of numbers Sum of original 6 numbers = 20 × 6 = 120 Sum of remaining 5 numbers = 15 × 5 = 75 Number removed = sum of original 6 numbers – sum of remaining 5 numbers

Number removed = 120 – 75 = 45

The removed number is 45

Example: 10 students of a class had a mean score of 70. The remaining 15 students of the class had mean score of 80. What is the mean score of the entire class?

Mean Word Problems

Example 1: Pedro’s luncheonette is open six days a week. His income for the first five days was $1,200, $1,100, $2,000, $1,400 and $3,000. How much money must she make on the sixth day to average $2,000 for the six days?

Example 2: George’s scores on three math tests were 70, 90 and 75. What score does he need on the fourth test to have a final average of 80?

How To Solve Word Problems With Averages?

• Timothy’s average score on the first four tests was 76. On his next five tests his average score was 85. What was his average score on all nine tests?
• Tracy mowed lawns for two hours and earned $7.40 an hour. Then she washed windows for three hours at $6.50 per hour. What was Tracy’s average earnings per hour for all five hours?
• After taking three quizzes, your average is 72 out of 100. What must your average be for the next five quizzes to increase your average to 77?

Solving Average Word Problems

Example: On her first four games, Jennifer bowled 101, 112, 126, 108. What is the minimum score she must bowl in her fifth game in order to have an average of 110?

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What is problem-solving and how to do it right steps, processes, exercises.

The better your problem-solving skills are, the better (and easier!) your life will be. Organized problem-solving is a killer career skill - learn all about it here.

Whether we’re trying to solve a technical problem at work, or trying to navigate around a roadblock that Google Maps doesn’t see – most people are problem-solving every single day . 

But how effective are you at tackling the challenges in your life? Do you have a bullet-proof process you follow that ensures solid outcomes, or... Do you act on a whim of inspiration (or lack thereof) to resolve your pressing problems?

Here’s the thing: the better your problem-solving skills are - the better (and easier!) your life will be (both professionally and personally). Organized problem-solving is a killer career (and life!) skill, so if you want to learn how to do it in the most efficient way possible, you’ve come to the right place.  

Read along to learn more about the steps, techniques and exercises of the problem-solving process.

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What is Problem-Solving?

We’re faced with the reality of having to solve problems every day, both in our private and professional lives. So why do we even need to learn about problem-solving? Aren’t we versed in it well enough already?

Well, what separates problem-solving from dealing with the usual day-to-day issues is that it’s a distinct process that allows you to go beyond the standard approaches to solving a problem and allows you to come up with more effective and efficient solutions. Or in other words, problem-solving allows you to knock out those problems with less effort. 

Just like with any other skill, there’s an efficient way to solve problems, and a non-efficient one. While it might be tempting to go for the quickest fix for your challenge without giving it much thought, it will only end up costing you more time down the road. Quick fixes are rarely (if ever!) effective and end up being massive time wasters. 

What separates problem-solving from dealing with the usual day-to-day issues is that it’s a distinct process that allows you to go beyond the standard approaches to solving a problem and allows you to come up with more effective and efficient solutions.

On the other hand, following a systemized clear process for problem-solving allows you to shortcut inefficiencies and time-wasters, turn your challenges into opportunities, and tackle problems of any scope without the usual stress and hassle. 

What is the process that you need to follow, then? We’re glad you asked...

The Five Stages of Problem-Solving

So what’s the best way to move through the problem-solving process? There’s a 5-step process that you can follow that will allow you to solve your challenges more efficiently and effectively. In short, you need to move through these 5 steps: 

• Defining a problem
• Ideating on a solution
• Committing to a course of action
• Implementing your solution
• And finally – analyzing the results. 

Let’s look at each of those stages in detail.

Step 1: Defining The Problem

The first step might sound obvious, but trust us, you don’t want to skip it! Clearly defining and framing your challenge will help you guide your efforts and make sure you’re focussing on the things that matter, instead of being distracted by a myriad of other options, problems and issues that come up. 

For once, you have to make sure you’re trying to solve the root cause, and not trying to mend the symptoms of it. For instance, if you keep losing users during your app onboarding process, you might jump to the conclusion that you need to tweak the process itself: change the copy, the screens, or the sequence of steps.

But unless you have clear evidence that confirms your hypothesis, your challenge might have an entirely different root cause, e.g. in confusing marketing communication prior to the app download. 

Clearly defining and framing your challenge will help you guide your efforts and make sure you’re focussing on the things that matter, all the while ensuring that you’re trying to solve the root cause, and not trying to mend the symptoms of it

That’s why it’s essential you take a close look at the entire problem, not just at a fraction of it.

There are several exercises that can help you get a broader, more holistic view of the problem, some of our all-time favorites include Expert Interviews, How Might We, or The Map. Check out the step-by-step instructions on how to run them (along with 5 more exercises for framing your challenge!) here. 

When in doubt, map out your challenge, and always try to tackle the bottlenecks that are more upstream - it’s likely that solving them will solve a couple of other challenges down the flow.

You also have to be mindful of how you frame the challenge: resist the urge to include a pre-defined solution into your problem statement. Priming your solutions to a predestined outcome destroys the purpose of following a step-by-step process in the first place!  

Steer clear of formulations like:

We need to change the onboarding process... or We need to improve ad copy to increase conversions. 

Instead, opt for more neutral, problem-oriented statements that don’t include a solution suggestion in them:

The drop off rate during the onboarding process is too high or Our ad conversion rates are below the norm.

Pro tip: Reframing your challenge as a ‘How Might We’ statement is a great way to spark up new ideas, opening your problem to a broader set of solutions, and is just a great way to reframe your problem into a more positive statement (without implying the possible solution!)

For example, following the onboarding drop-off rate problem we mentioned earlier, instead of framing it as a problem, you could opt for:

How Might We decrease the drop-off rate during the onboarding process? 

Find out more about the best exercises for problem framing here!

Now that you have a clear idea of what you’re trying to solve, it’s move on to the next phase of the problem-solving process.

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Step 2: ideating a solution.

Get ready to roll up your sleeves and challenge the status quo! This step of the problem-solving process is all about thinking outside of the box, challenging old assumptions, and thinking laterally. 

This stage is the one that tends to cause the most overwhelm in teams because it requires just the right balance of creativity and critical thinking, which tends to cause a lot of friction.

Our best advice?

Let go of the pressure to produce a polished, thought-through solution at this stage. You can hash out the details at a later point. Our goal right now is to come up with a direction, a prototype if you may, of where we want to move towards. 

Embrace the “quantity over quality” motto, and let your creative juices flow! Now, we’re not saying you should roll with sub-par ideas. But you shouldn’t get too fixated on feasibility and viability just yet . 

Your main goal during this step is to spark ideas, kick off your thinking process in the right direction, venture out of the familiar territories and think outside the box. 

For the ideation to be the most effective your team will have to feel safe to challenge the norm and wide-spread assumptions. So lay judgment by side, there is no space for “that’s the way it’s always been done” in this step.

For your ideation sessions to be as efficient as possible, we highly recommend to run them in a workshop setting: this helps reduce the usual drawbacks of open discussions in teams (i.e. groupthink & team politics!)

Our favorite exercises to run during this phase include Lightning Demos, Sketching, and variations of Brainstorming.  We crafted an entire article on how to run and facilitate these exercises in a separate article, so check it out of you’re going to be running an ideation session anytime soon!

Step 3: Choosing the Best Strategy & Committing

It’s time to decide which of the ideas that you generated in the last step will be the one you’ll implement. 

This step is arguably the hardest one to complete smoothly: groupthink, team politics, differences in opinions and communication styles all make it very hard to align a team on a common course of action. 

If you want to avoid the usual pitfalls of team decision-making, we recommend you steer clear of open unstructured discussion. While it’s useful in some scenarios, it’s a poor choice for when you need to make a decision, because it tends to reward the loudest people in the room, rather than give way to the best ideas. 

It’s crucial you not only commit to a course of action but get full buy-in from the team. If your team members don’t understand the reasons for a decision, or are not fully onboard, the implementation of your decision will be half-hearted, and that’s definitely not what you want! 

To achieve that, opt for anonymized, multi-layered voting, and include guided exercises like Storyboarding to prioritize your ideas. 

We’ve gathered the list of our top-rated decision-making exercises, along with step-by-step instructions on how to run them in this article!

As a bonus tip, we recommend you involve a facilitator throughout the entire process. They will help align the team, and guide them through prioritizing and de-prioritizing solutions, as well as defining the next steps. 

Pro tip : If you’re not the ultimate decision maker on the issue you’re trying to solve, make sure they’re in the room when the call is being made! Having a Decider in the room ensures that the decisions you come to will actually get executed on after, instead of getting shut down by your superiors after. 

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Step 4: implementing your solution.

Here’s a truth that might be hard to swallow: it doesn’t matter how innovative, creative, or original your idea is, if your execution is weak. 

One of our favourite illustrations of how this works in practice comes from the book “ Anything you want ” by Derek Sivers. He reveals that ideas should be treated as multipliers of execution. What this means is that a mediocre, “so-so” idea could be worth millions if executed well, while a “brilliant” idea can completely flop with bad execution. 

That’s why this step is crucial if you want to really master the problem-solving process. 

What do we mean by execution? Everything that happens after the whiteboards are wiped clean and your team starts to action the outcomes of your sessions, be it prototyping, development, or promotion. 

But don’t just take our word for it, look at the example of how execution affected Nintendo’s sales:

In the past few years, Nintendo has come up with 3 products: the Wii, the Wii U and the Switch. Check out their sales figures on the graph below - Wii is the clear-cut leader, followed by Switch, and finally Wii U lagging behind.

The Wii was unbelievably successful - it was a genuinely unique, “brilliant”-level idea and it had a “brilliant” execution (20x $10 million = $200 million). It is  one of the fastest selling game consoles of all time and it completely took over the market.

The next product was called Wii U and it was a “great” concept but the execution was absolutely terrible. So even though this product was very interesting and innovative, the end result was 15x $1,000 = $15,000. 

Finally, Nintendo took the Wii U concept and tried it again with the Switch. The idea was “so so” as it was already done before, but the execution was “brilliant”. So, 5x $10 million = $50 million! Much better.

Bottom line?  

The same idea can either make no dent in the market and damage your share price OR become a market hit and increase your share price dramatically. The only difference between the two scenarios – execution.

So shift your focus from coming up with crazy, innovative, outlandish ideas that will disrupt the market, and concentrate on really nailing down your execution instead. 

This is likely the least “workshoppy” step out of the entire problem-solving process because it requires less alignment and decision-making and more..well.. Execution!

But hey, we wouldn’t be called “Workshopper” if we didn't offer you at least one way to optimize and workshopify (yup, we’re making it a thing) your execution process. 

Cue in….prototyping. 

We’re huge fans of prototyping all big solutions (and testing them!) The main reason?

This saves us time AND money! Prototyping and testing your solutions (especially if they’re time and investment-demanding) is a great way to make sure you’re creating something that is actually needed. 

The key with prototyping the right way is to keep it simple. Don’t invest too much time, or resources into it. The goal is to gather data for your future decisions, not to create a near-to-perfect mockup of your solution.  

There are LOADS of prototyping forms and techniques, and if you’d like to learn more on the subject you should definitely check out our extensive prototyping guide.  

Step 5: Analyzing the Results

You’re nearly done, woo! Now that you have defined the right problem to tackle, brainstormed the solutions, aligned your team on the course of action, and put your plan into action it’s time to take stock of your efforts. 

Seek feedback from all involved parties, analyze the data you’ve gathered, look at the bottom line of your efforts, and  take a hard look at your problem: did it get solved? And even more than that, did the process feel smoother, easier, and more efficient than it normally is?

Running a retrospective is a great way to highlight things that went well and that you should keep for your next round of problem.solving, as well as pinpoint inefficiencies that you can eliminate.

‍ But which kind of retrospective should you run? There are loads of options, and it’s easy to feel overwhelmed by them all, so we gathered our favorite retrospective variations in this article.

And there you have it, you just completed the cycle of  problem-solving. We highly recommend you follow through with all the steps, without leaving any out. They all complement and build on each other, and it’s the combination of all 5 of them that makes the process effective. 

Now that you have the problem solving process down, you might be wondering…

Do I need any special skills in order to be able to move through that process?

And the answer is… sort of! More in this in the next section.

Problem-Solving Skills 

While your skill set will need to adapt and change based on the challenges you’ll be working on, most efficient problem-solvers have a solid foundation of these key skills:   

• Active listening. While you might be the expert in the area of your challenge, there’s not a single person on Earth that knows it all! Being open to others’ perspectives and practicing active listening will come in very handy during step 1 of the process, as you’re trying to define the scope and the exact angle of the problem you’re working on.
• Analytical approach. Your analytical skills will help you understand problems and effectively develop solutions. You will also need analytical skills during research to help distinguish between effective and ineffective solutions.
• Communication. Is there a single area of expertise that DOESN’T require strong communication skills? We honestly don’t think so! Just like with any other life area, clear communication can make or break your problem-solving process. Being able to clearly communicate why you need to solve this challenge to your team, as well as align your team on the course of action are crucial for the success of the process. 
• Decision-making. Ultimately, you will need to make a decision about how to solve problems that arise. A process without outcomes–regardless of how well thought-out and elaborate–is useless! If you want your problem-solving huddles to be effective, you have to come to grips with prioritization techniques and decision-making frameworks. 
• Facilitation. Problem-solving revolves around being able to guide a group or a team to a common decision, and facilitation skills are essential in making that happen. Knowing how to facilitate will make it easy to keep the group focussed on the challenge, shortcut circular discussions, and make sure you’re moving along to solving the problem instead of just treading waters with fruitless discussions. 

Not checking every single skill of your list just yet? Not to worry, the next section will give you practical tools on how to level up and improve your problem-solving skills.

How to Improve Your Problem-Solving Skills

Just like with any other skill, problem-solving is not an innate talent that you either have or you don’t.  There are concrete steps you can take to improve your skills. 

Here are some things that will get you closer to mastering the problem-solving process:

• Practice, Practice, Practice

Practice makes perfect, and problem-solving skills are no exception! Seek opportunities to utilize and develop these skills any time you can. 

If you don’t know where or how to start just yet, here’s a suggestion that will get you up and running in no time: run a quick problem-solving session on a challenge that has been bothering your team for a while now. 

It doesn’t need to be the big strategic decision or the issue defining the future of the company. Something easy and manageable (like optimizing office space or improving team communication) will do. 

As you start feeling more comfortable with the problem-solving techniques, you can start tackling bigger challenges. Before you know it, you’ll master the art of creative problem-solving!

• Use a tried and tested problem-solving workshop

Facilitation is one of the essential skills for problem-solving. But here’s the thing… Facilitation skills on their own won’t lead you to a solved challenge.

While being able to shortcut aimless discussions is a great skill, you have to make sure your problem-solving session has tangible outcomes. Using a tried and tested method, a workshop, is one of the easiest ways to do that. 

Our best advice is to get started with a tried and tested problem-solving workshop like the Lightning Decision Jam . The LDJ has all the right ingredients for quick, effective problem solving that leads to tangible outcomes. Give it a go!

• Learn from your peers

You may have colleagues who are skilled problem solvers. Observing how those colleagues solve problems can help you improve your own skills. 

If possible, ask one of your more experienced colleagues if you can observe their techniques. Ask them relevant questions and try to apply as many of the new found skills i your career as possible. 

• Learn & Practice the best problem-solving exercises

Having a toolbox of problem-solving exercises to pull from that can fit any type of challenge will make you a more versatile problem-solver and will make solving challenges that much easier for you! 

Once you get used to the groove of learning how to combine them into effective sessions or workshops, there’ll be no stopping you. What are some of the most effective problem-solving exercises? Glad you asked! We’ve gathered our favorite ones here, check it out! 

And there you have it, you’re now fully equipped for running creative problem-sessions with confidence and ease! Whichever method or exercise you choose, remember to keep track of your wins, and learn as much as you can from your losses! 

Anastasia Ushakova

Brand Strategist, Digital Marketer, and a Workshopper.

When Do You Need a Facilitator?

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• Math Article

Mean Questions

In mathematics and statistics, the mean is one of the important concepts that can be understood easily without much prior knowledge. Different types of the mean are defined based on the data provided. In this article, you will get questions and solutions for finding the mean of a given data set and grouped data (frequency distribution). These questions will help the students of Classes 9 and 10 to get enough practice for this concept.

What is Mean in Statistics?

Mean is one of the measures of central tendency in statistics. The mean is the average of the given data set, which means it can be calculated by dividing the sum of the given data values by the total number of data values.

Mean for ungrouped data:

Mean (x̄) = ∑x i /n

x i = x 1 , x 2 , x 3 ,…, x n such that i = 1, 2, 3,…n.

Number of observations = n

Mean for grouped data:

Mean (x̄) = ∑f i x i / ∑f i

Here, f i ’s are frequencies of x i ’s.

Mean Questions and Answers

1. Calculate the mean from the data showing marks of students in a class in a test: 40, 50, 55, 78, 58.

Given marks:

40, 50, 55, 78, 58

Here, the number of data values = 5

We know that:

Mean = Sum of data values/Total number of data values

= (40 + 50 + 55 + 78 + 58)/5

Therefore, the mean for the given data is 56.2.

2. A class consists of 50 students, out of which 30 are girls. The mean of marks scored by girls in a test is 73 (out of 100), and that of boys is 71. Determine the mean score of the whole class.

Total number of students in a class = 50

Number of girls in the class = 30

Number of boys in the class = 50 – 30 = 20

Mean marks scored by girls = 73

Mean marks scored by boys = 71

Thus, the total marks scored by girls = 73 × 30 = 2190

Also, the total marks scored by boys = 71 × 20 = 1420

Mean score of the class = (Total marks scored by girls and boys)/Total number of students

= (2190 + 1420)/50

3. The mean of the following distribution is 50.

Find the value of a and hence the frequencies of 30 and 70.

We know that,

Mean for grouped data (x̄) = ∑f i x i /n

(640a + 2800)/ (12a + 60) = 50 {given that mean = 50}

640a + 2800 = 50(12a + 60)

640a + 2800 = 600a + 3000

640a – 600a = 3000 – 2800

Therefore, the frequency for 30 = 5a + 3 = 5(5) + 3 = 28

And the frequency for 70 = 7a – 11 = 7(5) – 11 = 24

4. Find the mean salary of 60 workers of a factory from the following table:

Mean = (305000)/60 = 5083.33.

Therefore, the mean salary = Rs. 5083.33

5. A total of 25 patients admitted to a hospital are tested for levels of blood sugar, (mg/dl) and the results obtained were as follows:

87, 71, 83, 67, 85, 77, 69, 76, 65, 85, 85, 54, 70, 68, 80, 73, 78, 68, 85, 73, 81, 78, 81, 77, 75

Find the mean (mg/dl) of the above data.

Sum of data values = 87 + 71 + 83 + 67 + 85 + 77 + 69 + 76 + 65 + 85 + 85 + 54 + 70 + 68 + 80 + 73 + 78 + 68 + 85 + 73 + 81 + 78 + 81 + 77 + 75

Mean = 1891/25

6. Calculate the mean for the following distribution.

Mean = a + (∑f i d i /∑f i )

= 47.5 + (435/30)

= 47.5 + 14.5

7. The distribution below shows the number of wickets taken by bowlers in one-day cricket matches. Find the mean number of wickets by choosing a suitable method. What does the mean signify?

The given distribution has unequal class heights. So, we can use the direct method for calculating the mean.

Mean = ∑f i x i / ∑f i

Here the mean signifies that, on average, 45 bowlers take 152.89 wickets.

8. A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.

Which method did you use for finding the mean, and why?

Here, the values of fi and xi are small. Thus, we can use the direct method to calculate the mean.

So, Mean = ∑f i x i / ∑f i

Therefore, the mean number of plants = 8.1

9. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs 18. Find the missing frequency x.

Mean pocket allowance = a + (∑f i d i / ∑f i )

18 + [(2x – 40)/(44 + x)] = 18 [from the given]

(2x – 40)/(44 + x) = 18 – 18

(2x – 40)/(44 + x) = 0

2x – 40 = 0

Hence, the missing frequency is 20.

10. The following is the cumulative frequency distribution (of less than type) of 1000 persons, each one of age 20 years and above. Determine the mean age.

For the given frequency distribution, we need to write the class intervals and frequencies and then proceed with the necessary calculations for estimating the mean age.

Mean = a + h(∑f i u i /∑f i )

= 55 + 10(-370/1000)

= 55 – 3.7

Therefore, the mean age is 51.3 years.

Practice Questions on Mean

• The points scored by a basketball team in a series of matches are as follows: 17, 2, 7, 27, 25, 5, 14, 18, 10, 24, 48, 10, 8, 7, 10, 28

Find the mean for the data.

• 5 people were asked about the time in a week they spent doing social work in their community. They said 10, 7, 13, 20 and 15 hours, respectively. Find the mean (or average) time in a week devoted by them to social work.
• The following table gives the daily income of ten workers in a factory. Find the arithmetic mean.
• The frequency distribution table of agricultural holdings in a village is given below.

Find the mean agricultural holdings of the village.

• Calculate the mean for the following distribution.

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What Is Creative Problem-Solving & Why Is It Important?

• 01 Feb 2022

One of the biggest hindrances to innovation is complacency—it can be more comfortable to do what you know than venture into the unknown. Business leaders can overcome this barrier by mobilizing creative team members and providing space to innovate.

There are several tools you can use to encourage creativity in the workplace. Creative problem-solving is one of them, which facilitates the development of innovative solutions to difficult problems.

Here’s an overview of creative problem-solving and why it’s important in business.

Access your free e-book today.

What Is Creative Problem-Solving?

Research is necessary when solving a problem. But there are situations where a problem’s specific cause is difficult to pinpoint. This can occur when there’s not enough time to narrow down the problem’s source or there are differing opinions about its root cause.

In such cases, you can use creative problem-solving , which allows you to explore potential solutions regardless of whether a problem has been defined.

Creative problem-solving is less structured than other innovation processes and encourages exploring open-ended solutions. It also focuses on developing new perspectives and fostering creativity in the workplace . Its benefits include:

• Finding creative solutions to complex problems : User research can insufficiently illustrate a situation’s complexity. While other innovation processes rely on this information, creative problem-solving can yield solutions without it.
• Adapting to change : Business is constantly changing, and business leaders need to adapt. Creative problem-solving helps overcome unforeseen challenges and find solutions to unconventional problems.
• Fueling innovation and growth : In addition to solutions, creative problem-solving can spark innovative ideas that drive company growth. These ideas can lead to new product lines, services, or a modified operations structure that improves efficiency.

Creative problem-solving is traditionally based on the following key principles :

1. Balance Divergent and Convergent Thinking

Creative problem-solving uses two primary tools to find solutions: divergence and convergence. Divergence generates ideas in response to a problem, while convergence narrows them down to a shortlist. It balances these two practices and turns ideas into concrete solutions.

2. Reframe Problems as Questions

By framing problems as questions, you shift from focusing on obstacles to solutions. This provides the freedom to brainstorm potential ideas.

3. Defer Judgment of Ideas

When brainstorming, it can be natural to reject or accept ideas right away. Yet, immediate judgments interfere with the idea generation process. Even ideas that seem implausible can turn into outstanding innovations upon further exploration and development.

4. Focus on "Yes, And" Instead of "No, But"

Using negative words like "no" discourages creative thinking. Instead, use positive language to build and maintain an environment that fosters the development of creative and innovative ideas.

Creative Problem-Solving and Design Thinking

Whereas creative problem-solving facilitates developing innovative ideas through a less structured workflow, design thinking takes a far more organized approach.

Design thinking is a human-centered, solutions-based process that fosters the ideation and development of solutions. In the online course Design Thinking and Innovation , Harvard Business School Dean Srikant Datar leverages a four-phase framework to explain design thinking.

The four stages are:

• Clarify: The clarification stage allows you to empathize with the user and identify problems. Observations and insights are informed by thorough research. Findings are then reframed as problem statements or questions.
• Ideate: Ideation is the process of coming up with innovative ideas. The divergence of ideas involved with creative problem-solving is a major focus.
• Develop: In the development stage, ideas evolve into experiments and tests. Ideas converge and are explored through prototyping and open critique.
• Implement: Implementation involves continuing to test and experiment to refine the solution and encourage its adoption.

Creative problem-solving primarily operates in the ideate phase of design thinking but can be applied to others. This is because design thinking is an iterative process that moves between the stages as ideas are generated and pursued. This is normal and encouraged, as innovation requires exploring multiple ideas.

Creative Problem-Solving Tools

While there are many useful tools in the creative problem-solving process, here are three you should know:

Creating a Problem Story

One way to innovate is by creating a story about a problem to understand how it affects users and what solutions best fit their needs. Here are the steps you need to take to use this tool properly.

1. Identify a UDP

Create a problem story to identify the undesired phenomena (UDP). For example, consider a company that produces printers that overheat. In this case, the UDP is "our printers overheat."

2. Move Forward in Time

To move forward in time, ask: “Why is this a problem?” For example, minor damage could be one result of the machines overheating. In more extreme cases, printers may catch fire. Don't be afraid to create multiple problem stories if you think of more than one UDP.

3. Move Backward in Time

To move backward in time, ask: “What caused this UDP?” If you can't identify the root problem, think about what typically causes the UDP to occur. For the overheating printers, overuse could be a cause.

Following the three-step framework above helps illustrate a clear problem story:

• The printer is overused.
• The printer overheats.
• The printer breaks down.

You can extend the problem story in either direction if you think of additional cause-and-effect relationships.

4. Break the Chains

By this point, you’ll have multiple UDP storylines. Take two that are similar and focus on breaking the chains connecting them. This can be accomplished through inversion or neutralization.

• Inversion: Inversion changes the relationship between two UDPs so the cause is the same but the effect is the opposite. For example, if the UDP is "the more X happens, the more likely Y is to happen," inversion changes the equation to "the more X happens, the less likely Y is to happen." Using the printer example, inversion would consider: "What if the more a printer is used, the less likely it’s going to overheat?" Innovation requires an open mind. Just because a solution initially seems unlikely doesn't mean it can't be pursued further or spark additional ideas.
• Neutralization: Neutralization completely eliminates the cause-and-effect relationship between X and Y. This changes the above equation to "the more or less X happens has no effect on Y." In the case of the printers, neutralization would rephrase the relationship to "the more or less a printer is used has no effect on whether it overheats."

Even if creating a problem story doesn't provide a solution, it can offer useful context to users’ problems and additional ideas to be explored. Given that divergence is one of the fundamental practices of creative problem-solving, it’s a good idea to incorporate it into each tool you use.

Brainstorming

Brainstorming is a tool that can be highly effective when guided by the iterative qualities of the design thinking process. It involves openly discussing and debating ideas and topics in a group setting. This facilitates idea generation and exploration as different team members consider the same concept from multiple perspectives.

Hosting brainstorming sessions can result in problems, such as groupthink or social loafing. To combat this, leverage a three-step brainstorming method involving divergence and convergence :

• Have each group member come up with as many ideas as possible and write them down to ensure the brainstorming session is productive.
• Continue the divergence of ideas by collectively sharing and exploring each idea as a group. The goal is to create a setting where new ideas are inspired by open discussion.
• Begin the convergence of ideas by narrowing them down to a few explorable options. There’s no "right number of ideas." Don't be afraid to consider exploring all of them, as long as you have the resources to do so.

Alternate Worlds

The alternate worlds tool is an empathetic approach to creative problem-solving. It encourages you to consider how someone in another world would approach your situation.

For example, if you’re concerned that the printers you produce overheat and catch fire, consider how a different industry would approach the problem. How would an automotive expert solve it? How would a firefighter?

Be creative as you consider and research alternate worlds. The purpose is not to nail down a solution right away but to continue the ideation process through diverging and exploring ideas.

Continue Developing Your Skills

Whether you’re an entrepreneur, marketer, or business leader, learning the ropes of design thinking can be an effective way to build your skills and foster creativity and innovation in any setting.

If you're ready to develop your design thinking and creative problem-solving skills, explore Design Thinking and Innovation , one of our online entrepreneurship and innovation courses. If you aren't sure which course is the right fit, download our free course flowchart to determine which best aligns with your goals.

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