There is the misuse of the terms of statistics and probability in many domains. Politics being one. The #Noestimates advocates are another of the most prolific abusers of these terms. Here are the mathematical definitions. not the Wikipedia definition, not the self-made definitions used to support their conjectures.
- An Estimate is a value inferred for a population of values based on data collected from a sample of data from that population. The estimate can also be produced parametrically or through a simulation (Monte Carlo is common, but Method of Moments is another we use).
- Estimates can be about the past, present, or future.
- We can estimate the number of clams in the Pleistocene era that are in the shale formations near our house.
- We can estimate the number of people sitting in Folsom Field for lat night's game against Utah. The Buff's won and are now the PAC-12 South Champs.
- We can estimate the total cost, total duration, and the probability that all the Features will be delivered on the program we are working for the US Government. Or ANY software project for that matter.
- Estimates have precision and accuracy.
- These values are estimates as well.
- The estimated completion cost for this program is $357,000,000 with an accuracy of $200,000 and a precision of $300,000.
- Another way to speak about the estimated cost is This program will cost $357,000,000 or less with 80% confidence.
- An estimate is a statistic about a whole population of possible values from a previous reference period or a model that can generate possible values given the conditions of the model.
An estimate is the calculated approximation of a result
- Forecasts speculate future values for a population of possible values with a certain level of confidence, based on the current and past values as an expectation (prediction) of what will happen:
- This is the basis of weather forecasting.
- If you listen carefully to the weather forecast it says there is a 30% chance of snow next week over the forecast area.
- We live at the mouth of a canyon at 5,095' and of there is a 30% chance of snow in Boulder (8 miles south), there is a much lower chance in our neighborhood.
- Business forecasts, weather forecasts, traffic forecasts are typical. These forecasts usually come from models of the process being forecast. NOAA and NCAR are in our town, so lots of forecasting going on. Weather as well as climate.
- Not so typical to Forecast the cost of a project or forecast the delivery date. Those are estimated values.
- For example a financial statement presents, to the best of the responsible party's knowledge and belief, an entity's expected financial position, results of operations, and cash flows. 
- In a forecast, the assumptions represent expectations of actual future events.
- Sales forecasts
- Revenue growth
- Weather forecasts
- Forecasts of cattle prices in the spring
A forecast is a prediction of some out come in the future. Forecasts are based on estimating the processes that produce the forecast. The underlying statistcal models (circulation, thermal models) of weather forecasting are estimates of the compressable fluid flow of gases and moisture in the atmsophere (way over simplified).
- Projections indicate what future values may exist for a population of values if the assumed patterns of change were to occur. Projections are not a prediction that the population will change in that manner.
- Projected revenue for GE aircraft engine sales in 2017 was an article in this week's Aviation Week & Space Technology.
- A projection simply indicates a future value for the population if the set of underlying assumptions occurs.
A prediction says something about the future.
Project cost, schedule, and technical performance Estimates
All projects contain uncertainty. Uncertainty comes in two forms - aleatory (irreducible) and epistemic (reducible). If we're going to make decisions in the presence of these uncertainties, we need to estimate what their values are, what the range of values are, what the stability of the underlying processes that generate these values are, how these values interact with all the elements of the project and what the impact of these ranges of value will do to the probability of success of our project.
Project decision in the presence of unceratinty cannot be made without estimates. Anyomne claiming othewise does not understand statsiics and probability of orucomes on projects
As well anyone claims estimates are a waste, not needed, misused by management, of any other dysfunction, is doing them wrong. So as we said at Rocky Flats - Don't Do Stupid Things On Purpose. Which means when you do hear those phrases, you'll know they are Doing Stupid Things on Purpose.
And when yo hear, we don't need estimates we need budget, remember:
In a world of limited funds, as a project manager, Product Owner, or even sole contributor, you’re constantly deciding how to get the most return for your investment. The more accurate your estimate of project cost is, the better able you will be to manage your project’s budget.
Another example of not understanding the probability and statistics of projects and the businesses that fund them is, there are two estimates needed for all projects that operate in the presence of uncertainty:
- Estimate at Completion (EAC)
- EAC is the expected cost of the project when it is complete.
- This can be calculated bottom-up from the past performance and future projections of performance for the projects work - which in the future will be an estimate.
- Estimate to Complete
- ETC is the expected cost to complete the project.
- The ETC used to calculate the EAC
- EAC = Actual Costs to Date (AC) + Estimated Cost to Complete (ETC).
- EAC = Actual performance to date / Some Index of Performance.
This last formula is universal and can be used no matter the software development method.
- Agile has such a formula - it's called the Burn Down Chart. We're burning down story points at some rate. If we continue at this rate, we will be done by this estimated date
- Same for traditional projects. We're spending at a current rate - the run rate - if we keep spending at this rate, the project will cost that much
- Earned Value Management provides the same EAC and can also provide an Estimated Completion Date (ECD)
- Earned Schedule provide a better ECD
Nothing in progression can rest on its orginal plan - Thomas Monson 
All project work is driven by uncertainty. Uncertainty is modeled by random variables. These variables can represent aleatory uncertainty or epistemic uncertainty. This uncertainty creates risk and as stated here often
Risk Management is How Adults Mange Projects - Time Lister
So if you hear the conjecture that decisioins can be made in the presence of uncertainty without estimates, you'll now know that is a complete load a crap, run away.
If this topic interests you here's a Bibliography of materials for estimating and lots of other topics in agile software development that is updated all the time. Please read and use these when you hear unsubstantiated claims around estimating in the presnece of uncertainty. Making estimates is our business and this resoruce has served us well over the decades.
- Australian Bureau of Statistics
- Financial Forecasts and Projections, AT §301.06, AICPA, 2015
- Earned Value Management in EIA-748-C
- Earned Schedule uses the same values as Earned Value, to produce an estimated complete date, www.earnedschedule.com I started using ES at Rocky Flats to explain to the steel workers that their productivity as measured in Budgeted Cost for Work Complete (BCWP or EV) means they are late. ES told them how late and provides the date of the projected completion of the planned work
- Project Management Analytics: A Data-Driven Approach to Making Rational and Effective Project Decisions, Harjit Singh, Pearson FT Press; 1st Edition, November 12, 2015.